Elementarmathematik Beispiele

$\cot (x) \sec^{4} (x) = \cot (x) + 2 \tan (x) + \tan^{3} (x)$

Schritt 1

Vereinfache die linke Seite.

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Schritt 1.1

Vereinfache $\cot (x) \sec^{4} (x)$ .

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Schritt 1.1.1

Schreibe $\cot (x)$ mithilfe von Sinus und Kosinus um.

$\frac{\cos (x)}{\sin (x)} \sec^{4} (x) = \cot (x) + 2 \tan (x) + \tan^{3} (x)$

Schritt 1.1.2

Schreibe $\sec (x)$ mithilfe von Sinus und Kosinus um.

$\frac{\cos (x)}{\sin (x)} {(\frac{1}{\cos (x)})}^{4} = \cot (x) + 2 \tan (x) + \tan^{3} (x)$

Schritt 1.1.3

Vereinfache den Ausdruck.

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Schritt 1.1.3.1

Wende die Produktregel auf $\frac{1}{\cos (x)}$ an.

$\frac{\cos (x)}{\sin (x)} \cdot \frac{1^{4}}{\cos^{4} (x)} = \cot (x) + 2 \tan (x) + \tan^{3} (x)$

Schritt 1.1.3.2

Eins zu einer beliebigen Potenz erhoben ergibt eins.

$\frac{\cos (x)}{\sin (x)} \cdot \frac{1}{\cos^{4} (x)} = \cot (x) + 2 \tan (x) + \tan^{3} (x)$

Schritt 1.1.4

Mutltipliziere $\frac{\cos (x)}{\sin (x)}$ mit $\frac{1}{\cos^{4} (x)}$ .

$\frac{\cos (x)}{\sin (x) \cos^{4} (x)} = \cot (x) + 2 \tan (x) + \tan^{3} (x)$

Schritt 1.1.5

Kürze den gemeinsamen Teiler von $\cos (x)$ und $\cos^{4} (x)$ .

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Schritt 1.1.5.1

Multipliziere mit $1$ .

$\frac{\cos (x) \cdot 1}{\sin (x) \cos^{4} (x)} = \cot (x) + 2 \tan (x) + \tan^{3} (x)$

Schritt 1.1.5.2

Kürze die gemeinsamen Faktoren.

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Schritt 1.1.5.2.1

Faktorisiere $\cos (x)$ aus $\sin (x) \cos^{4} (x)$ heraus.

$\frac{\cos (x) \cdot 1}{\cos (x) (\sin (x) \cos^{3} (x))} = \cot (x) + 2 \tan (x) + \tan^{3} (x)$

Schritt 1.1.5.2.2

Kürze den gemeinsamen Faktor.

$\frac{\cos (x) \cdot 1}{\cos (x) (\sin (x) \cos^{3} (x))} = \cot (x) + 2 \tan (x) + \tan^{3} (x)$

Schritt 1.1.5.2.3

Forme den Ausdruck um.

$\frac{1}{\sin (x) \cos^{3} (x)} = \cot (x) + 2 \tan (x) + \tan^{3} (x)$

Schritt 1.1.6

Separiere Brüche.

$\frac{1}{\cos^{3} (x)} \cdot \frac{1}{\sin (x)} = \cot (x) + 2 \tan (x) + \tan^{3} (x)$

Schritt 1.1.7

Wandle von $\frac{1}{\sin (x)}$ nach $\csc (x)$ um.

$\frac{1}{\cos^{3} (x)} \csc (x) = \cot (x) + 2 \tan (x) + \tan^{3} (x)$

Schritt 1.1.8

Kombiniere $\frac{1}{\cos^{3} (x)}$ und $\csc (x)$ .

$\frac{\csc (x)}{\cos^{3} (x)} = \cot (x) + 2 \tan (x) + \tan^{3} (x)$

Schritt 1.1.9

Faktorisiere $\cos (x)$ aus $\cos^{3} (x)$ heraus.

$\frac{\csc (x)}{\cos (x) \cos^{2} (x)} = \cot (x) + 2 \tan (x) + \tan^{3} (x)$

Schritt 1.1.10

Separiere Brüche.

$\frac{1}{\cos^{2} (x)} \cdot \frac{\csc (x)}{\cos (x)} = \cot (x) + 2 \tan (x) + \tan^{3} (x)$

Schritt 1.1.11

Schreibe $\csc (x)$ mithilfe von Sinus und Kosinus um.

$\frac{1}{\cos^{2} (x)} \cdot \frac{\frac{1}{\sin (x)}}{\cos (x)} = \cot (x) + 2 \tan (x) + \tan^{3} (x)$

Schritt 1.1.12

Schreibe $\frac{\frac{1}{\sin (x)}}{\cos (x)}$ als ein Produkt um.

$\frac{1}{\cos^{2} (x)} (\frac{1}{\sin (x)} \cdot \frac{1}{\cos (x)}) = \cot (x) + 2 \tan (x) + \tan^{3} (x)$

Schritt 1.1.13

Vereinfache.

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Schritt 1.1.13.1

Wandle von $\frac{1}{\sin (x)}$ nach $\csc (x)$ um.

$\frac{1}{\cos^{2} (x)} (\csc (x) \frac{1}{\cos (x)}) = \cot (x) + 2 \tan (x) + \tan^{3} (x)$

Schritt 1.1.13.2

Wandle von $\frac{1}{\cos (x)}$ nach $\sec (x)$ um.

$\frac{1}{\cos^{2} (x)} (\csc (x) \sec (x)) = \cot (x) + 2 \tan (x) + \tan^{3} (x)$

Schritt 1.1.14

Schreibe $1$ als $1^{2}$ um.

$\frac{1^{2}}{\cos^{2} (x)} (\csc (x) \sec (x)) = \cot (x) + 2 \tan (x) + \tan^{3} (x)$

Schritt 1.1.15

Schreibe $\frac{1^{2}}{\cos^{2} (x)}$ als ${(\frac{1}{\cos (x)})}^{2}$ um.

${(\frac{1}{\cos (x)})}^{2} (\csc (x) \sec (x)) = \cot (x) + 2 \tan (x) + \tan^{3} (x)$

Schritt 1.1.16

Wandle von $\frac{1}{\cos (x)}$ nach $\sec (x)$ um.

$\sec^{2} (x) (\csc (x) \sec (x)) = \cot (x) + 2 \tan (x) + \tan^{3} (x)$

Schritt 1.1.17

Multipliziere $\sec^{2} (x)$ mit $\sec (x)$ durch Addieren der Exponenten.

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Schritt 1.1.17.1

Bewege $\sec (x)$ .

$\sec (x) \sec^{2} (x) \csc (x) = \cot (x) + 2 \tan (x) + \tan^{3} (x)$

Schritt 1.1.17.2

Mutltipliziere $\sec (x)$ mit $\sec^{2} (x)$ .

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Schritt 1.1.17.2.1

Potenziere $\sec (x)$ mit $1$ .

$\sec^{1} (x) \sec^{2} (x) \csc (x) = \cot (x) + 2 \tan (x) + \tan^{3} (x)$

Schritt 1.1.17.2.2

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

${\sec (x)}^{1 + 2} \csc (x) = \cot (x) + 2 \tan (x) + \tan^{3} (x)$

Schritt 1.1.17.3

Addiere $1$ und $2$ .

$\sec^{3} (x) \csc (x) = \cot (x) + 2 \tan (x) + \tan^{3} (x)$

Schritt 2

Da $x$ auf der rechten Seite der Gleichung ist, vertausche die Seiten, sodass es auf der linken Seite ist.

$\cot (x) + 2 \tan (x) + \tan^{3} (x) = \sec^{3} (x) \csc (x)$

Schritt 3

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Schritt 3.1

Vereinfache jeden Term.

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Schritt 3.1.1

Schreibe $\cot (x)$ mithilfe von Sinus und Kosinus um.

$\frac{\cos (x)}{\sin (x)} + 2 \tan (x) + \tan^{3} (x) = \sec^{3} (x) \csc (x)$

Schritt 3.1.2

Schreibe $\tan (x)$ mithilfe von Sinus und Kosinus um.

$\frac{\cos (x)}{\sin (x)} + 2 \frac{\sin (x)}{\cos (x)} + \tan^{3} (x) = \sec^{3} (x) \csc (x)$

Schritt 3.1.3

Kombiniere $2$ und $\frac{\sin (x)}{\cos (x)}$ .

$\frac{\cos (x)}{\sin (x)} + \frac{2 \sin (x)}{\cos (x)} + \tan^{3} (x) = \sec^{3} (x) \csc (x)$

Schritt 3.1.4

Schreibe $\tan (x)$ mithilfe von Sinus und Kosinus um.

$\frac{\cos (x)}{\sin (x)} + \frac{2 \sin (x)}{\cos (x)} + {(\frac{\sin (x)}{\cos (x)})}^{3} = \sec^{3} (x) \csc (x)$

Schritt 3.1.5

Wende die Produktregel auf $\frac{\sin (x)}{\cos (x)}$ an.

$\frac{\cos (x)}{\sin (x)} + \frac{2 \sin (x)}{\cos (x)} + \frac{\sin^{3} (x)}{\cos^{3} (x)} = \sec^{3} (x) \csc (x)$

Schritt 4

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Schritt 4.1

Vereinfache $\sec^{3} (x) \csc (x)$ .

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Schritt 4.1.1

Schreibe $\sec (x)$ mithilfe von Sinus und Kosinus um.

$\frac{\cos (x)}{\sin (x)} + \frac{2 \sin (x)}{\cos (x)} + \frac{\sin^{3} (x)}{\cos^{3} (x)} = {(\frac{1}{\cos (x)})}^{3} \csc (x)$

Schritt 4.1.2

Wende die Produktregel auf $\frac{1}{\cos (x)}$ an.

$\frac{\cos (x)}{\sin (x)} + \frac{2 \sin (x)}{\cos (x)} + \frac{\sin^{3} (x)}{\cos^{3} (x)} = \frac{1^{3}}{\cos^{3} (x)} \csc (x)$

Schritt 4.1.3

Eins zu einer beliebigen Potenz erhoben ergibt eins.

$\frac{\cos (x)}{\sin (x)} + \frac{2 \sin (x)}{\cos (x)} + \frac{\sin^{3} (x)}{\cos^{3} (x)} = \frac{1}{\cos^{3} (x)} \csc (x)$

Schritt 4.1.4

Schreibe $\csc (x)$ mithilfe von Sinus und Kosinus um.

$\frac{\cos (x)}{\sin (x)} + \frac{2 \sin (x)}{\cos (x)} + \frac{\sin^{3} (x)}{\cos^{3} (x)} = \frac{1}{\cos^{3} (x)} \cdot \frac{1}{\sin (x)}$

Schritt 4.1.5

Kombinieren.

$\frac{\cos (x)}{\sin (x)} + \frac{2 \sin (x)}{\cos (x)} + \frac{\sin^{3} (x)}{\cos^{3} (x)} = \frac{1 \cdot 1}{\cos^{3} (x) \sin (x)}$

Schritt 4.1.6

Mutltipliziere $1$ mit $1$ .

$\frac{\cos (x)}{\sin (x)} + \frac{2 \sin (x)}{\cos (x)} + \frac{\sin^{3} (x)}{\cos^{3} (x)} = \frac{1}{\cos^{3} (x) \sin (x)}$

Schritt 5

Multipliziere beide Seiten der Gleichung mit $\sin (x)$ .

$\sin (x) (\frac{\cos (x)}{\sin (x)} + \frac{2 \sin (x)}{\cos (x)} + \frac{\sin^{3} (x)}{\cos^{3} (x)}) = \sin (x) \frac{1}{\cos^{3} (x) \sin (x)}$

Schritt 6

Wende das Distributivgesetz an.

$\sin (x) \frac{\cos (x)}{\sin (x)} + \sin (x) \frac{2 \sin (x)}{\cos (x)} + \sin (x) \frac{\sin^{3} (x)}{\cos^{3} (x)} = \sin (x) \frac{1}{\cos^{3} (x) \sin (x)}$

Schritt 7

Vereinfache.

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Schritt 7.1

Kürze den gemeinsamen Faktor von $\sin (x)$ .

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Schritt 7.1.1

Kürze den gemeinsamen Faktor.

$\sin (x) \frac{\cos (x)}{\sin (x)} + \sin (x) \frac{2 \sin (x)}{\cos (x)} + \sin (x) \frac{\sin^{3} (x)}{\cos^{3} (x)} = \sin (x) \frac{1}{\cos^{3} (x) \sin (x)}$

Schritt 7.1.2

Forme den Ausdruck um.

$\cos (x) + \sin (x) \frac{2 \sin (x)}{\cos (x)} + \sin (x) \frac{\sin^{3} (x)}{\cos^{3} (x)} = \sin (x) \frac{1}{\cos^{3} (x) \sin (x)}$

Schritt 7.2

Multipliziere $\sin (x) \frac{2 \sin (x)}{\cos (x)}$ .

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Schritt 7.2.1

Kombiniere $\sin (x)$ und $\frac{2 \sin (x)}{\cos (x)}$ .

$\cos (x) + \frac{\sin (x) (2 \sin (x))}{\cos (x)} + \sin (x) \frac{\sin^{3} (x)}{\cos^{3} (x)} = \sin (x) \frac{1}{\cos^{3} (x) \sin (x)}$

Schritt 7.2.2

Potenziere $\sin (x)$ mit $1$ .

$\cos (x) + \frac{2 (\sin^{1} (x) \sin (x))}{\cos (x)} + \sin (x) \frac{\sin^{3} (x)}{\cos^{3} (x)} = \sin (x) \frac{1}{\cos^{3} (x) \sin (x)}$

Schritt 7.2.3

Potenziere $\sin (x)$ mit $1$ .

$\cos (x) + \frac{2 (\sin^{1} (x) \sin^{1} (x))}{\cos (x)} + \sin (x) \frac{\sin^{3} (x)}{\cos^{3} (x)} = \sin (x) \frac{1}{\cos^{3} (x) \sin (x)}$

Schritt 7.2.4

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$\cos (x) + \frac{2 {\sin (x)}^{1 + 1}}{\cos (x)} + \sin (x) \frac{\sin^{3} (x)}{\cos^{3} (x)} = \sin (x) \frac{1}{\cos^{3} (x) \sin (x)}$

Schritt 7.2.5

Addiere $1$ und $1$ .

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \sin (x) \frac{\sin^{3} (x)}{\cos^{3} (x)} = \sin (x) \frac{1}{\cos^{3} (x) \sin (x)}$

Schritt 7.3

Multipliziere $\sin (x) \frac{\sin^{3} (x)}{\cos^{3} (x)}$ .

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Schritt 7.3.1

Kombiniere $\sin (x)$ und $\frac{\sin^{3} (x)}{\cos^{3} (x)}$ .

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin (x) \sin^{3} (x)}{\cos^{3} (x)} = \sin (x) \frac{1}{\cos^{3} (x) \sin (x)}$

Schritt 7.3.2

Multipliziere $\sin (x)$ mit $\sin^{3} (x)$ durch Addieren der Exponenten.

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Schritt 7.3.2.1

Mutltipliziere $\sin (x)$ mit $\sin^{3} (x)$ .

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Schritt 7.3.2.1.1

Potenziere $\sin (x)$ mit $1$ .

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{1} (x) \sin^{3} (x)}{\cos^{3} (x)} = \sin (x) \frac{1}{\cos^{3} (x) \sin (x)}$

Schritt 7.3.2.1.2

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{{\sin (x)}^{1 + 3}}{\cos^{3} (x)} = \sin (x) \frac{1}{\cos^{3} (x) \sin (x)}$

Schritt 7.3.2.2

Addiere $1$ und $3$ .

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} = \sin (x) \frac{1}{\cos^{3} (x) \sin (x)}$

Schritt 8

Kürze den gemeinsamen Faktor von $\sin (x)$ .

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Schritt 8.1

Faktorisiere $\sin (x)$ aus $\cos^{3} (x) \sin (x)$ heraus.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} = \sin (x) \frac{1}{\sin (x) \cos^{3} (x)}$

Schritt 8.2

Kürze den gemeinsamen Faktor.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} = \sin (x) \frac{1}{\sin (x) \cos^{3} (x)}$

Schritt 8.3

Forme den Ausdruck um.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} = \frac{1}{\cos^{3} (x)}$

Schritt 9

Subtrahiere $\frac{1}{\cos^{3} (x)}$ von beiden Seiten der Gleichung.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = 0$

Schritt 10

Vereinfache jeden Term.

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Schritt 10.1

Faktorisiere $\sin (x)$ aus $\sin^{2} (x)$ heraus.

$\cos (x) + \frac{2 (\sin (x) \sin (x))}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = 0$

Schritt 10.2

Separiere Brüche.

$\cos (x) + \frac{2 (\sin (x))}{1} \cdot \frac{\sin (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = 0$

Schritt 10.3

Wandle von $\frac{\sin (x)}{\cos (x)}$ nach $\tan (x)$ um.

$\cos (x) + \frac{2 (\sin (x))}{1} \tan (x) + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = 0$

Schritt 10.4

Dividiere $2 (\sin (x))$ durch $1$ .

$\cos (x) + 2 (\sin (x)) \tan (x) + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = 0$

Schritt 10.5

Faktorisiere $\sin^{3} (x)$ aus $\sin^{4} (x)$ heraus.

$\cos (x) + 2 \sin (x) \tan (x) + \frac{\sin^{3} (x) \sin (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = 0$

Schritt 10.6

Multipliziere mit $1$ .

$\cos (x) + 2 \sin (x) \tan (x) + \frac{\sin^{3} (x) \sin (x)}{\cos^{3} (x) \cdot 1} - \frac{1}{\cos^{3} (x)} = 0$

Schritt 10.7

Separiere Brüche.

$\cos (x) + 2 \sin (x) \tan (x) + \frac{\sin (x)}{1} \cdot \frac{\sin^{3} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = 0$

Schritt 10.8

Wandle von $\frac{\sin^{3} (x)}{\cos^{3} (x)}$ nach $\tan^{3} (x)$ um.

$\cos (x) + 2 \sin (x) \tan (x) + \frac{\sin (x)}{1} \tan^{3} (x) - \frac{1}{\cos^{3} (x)} = 0$

Schritt 10.9

Dividiere $\sin (x)$ durch $1$ .

$\cos (x) + 2 \sin (x) \tan (x) + \sin (x) \tan^{3} (x) - \frac{1}{\cos^{3} (x)} = 0$

Schritt 11

Teile jeden Term in der Gleichung durch $\cos (x)$ .

$\frac{\cos (x)}{\cos (x)} + \frac{2 \sin (x) \tan (x)}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 12

Kürze den gemeinsamen Faktor von $\cos (x)$ .

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Schritt 12.1

Kürze den gemeinsamen Faktor.

$\frac{\cos (x)}{\cos (x)} + \frac{2 \sin (x) \tan (x)}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 12.2

Forme den Ausdruck um.

$1 + \frac{2 \sin (x) \tan (x)}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 13

Vereinfache den Zähler.

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Schritt 13.1

Schreibe $\tan (x)$ mithilfe von Sinus und Kosinus um.

$1 + \frac{2 \sin (x) (\frac{\sin (x)}{\cos (x)})}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 13.2

Kombiniere Exponenten.

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Schritt 13.2.1

Kombiniere $\frac{\sin (x)}{\cos (x)}$ und $2$ .

$1 + \frac{\frac{\sin (x) \cdot 2}{\cos (x)} \cdot \sin (x)}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 13.2.2

Kombiniere $\frac{\sin (x) \cdot 2}{\cos (x)}$ und $\sin (x)$ .

$1 + \frac{\frac{\sin (x) \cdot (2 \sin (x))}{\cos (x)}}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 13.2.3

Potenziere $\sin (x)$ mit $1$ .

$1 + \frac{\frac{2 (\sin (x) \sin (x))}{\cos (x)}}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 13.2.4

Potenziere $\sin (x)$ mit $1$ .

$1 + \frac{\frac{2 (\sin (x) \sin (x))}{\cos (x)}}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 13.2.5

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$1 + \frac{\frac{2 {\sin (x)}^{1 + 1}}{\cos (x)}}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 13.2.6

Addiere $1$ und $1$ .

$1 + \frac{\frac{2 \sin^{2} (x)}{\cos (x)}}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 14

Multipliziere den Zähler mit dem Kehrwert des Nenners.

$1 + \frac{2 \sin^{2} (x)}{\cos (x)} \cdot \frac{1}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 15

Kombiniere Brüche.

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Schritt 15.1

Kombinieren.

$1 + \frac{2 \sin^{2} (x) \cdot 1}{\cos (x) \cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 15.2

Mutltipliziere $2$ mit $1$ .

$1 + \frac{2 \sin^{2} (x)}{\cos (x) \cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 16

Vereinfache den Nenner.

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Schritt 16.1

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$1 + \frac{2 \sin^{2} (x)}{{\cos (x)}^{1 + 1}} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 16.2

Addiere $1$ und $1$ .

$1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 17

Multipliziere mit $1$ .

$1 + \frac{2 (\sin^{2} (x) \cdot 1)}{\cos^{2} (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 18

Multipliziere mit $1$ .

$1 + \frac{2 (\sin^{2} (x) \cdot 1)}{\cos^{2} (x) \cdot 1} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 19

Separiere Brüche.

$1 + \frac{2 \cdot (1)}{1} \cdot \frac{\sin^{2} (x)}{\cos^{2} (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 20

Wandle von $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ nach $\tan^{2} (x)$ um.

$1 + \frac{2 \cdot (1)}{1} \cdot \tan^{2} (x) + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 21

Mutltipliziere $2$ mit $1$ .

$1 + \frac{2}{1} \cdot \tan^{2} (x) + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 22

Dividiere $2$ durch $1$ .

$1 + 2 \tan^{2} (x) + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 23

Vereinfache den Zähler.

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Schritt 23.1

Schreibe $\tan (x)$ mithilfe von Sinus und Kosinus um.

$1 + 2 \tan^{2} (x) + \frac{\sin (x) {(\frac{\sin (x)}{\cos (x)})}^{3}}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 23.2

Wende die Produktregel auf $\frac{\sin (x)}{\cos (x)}$ an.

$1 + 2 \tan^{2} (x) + \frac{\sin (x) (\frac{\sin^{3} (x)}{\cos^{3} (x)})}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 24

Kombiniere $\sin (x)$ und $\frac{\sin^{3} (x)}{\cos^{3} (x)}$ .

$1 + 2 \tan^{2} (x) + \frac{\frac{\sin (x) \sin^{3} (x)}{\cos^{3} (x)}}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 25

Multipliziere $\sin (x)$ mit $\sin^{3} (x)$ durch Addieren der Exponenten.

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Schritt 25.1

Mutltipliziere $\sin (x)$ mit $\sin^{3} (x)$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 25.1.1

Potenziere $\sin (x)$ mit $1$ .

$1 + 2 \tan^{2} (x) + \frac{\frac{\sin (x) \sin^{3} (x)}{\cos^{3} (x)}}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 25.1.2

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$1 + 2 \tan^{2} (x) + \frac{\frac{{\sin (x)}^{1 + 3}}{\cos^{3} (x)}}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 25.2

Addiere $1$ und $3$ .

$1 + 2 \tan^{2} (x) + \frac{\frac{\sin^{4} (x)}{\cos^{3} (x)}}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 26

Multipliziere den Zähler mit dem Kehrwert des Nenners.

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x)}{\cos^{3} (x)} \cdot \frac{1}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 27

Kombinieren.

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x) \cdot 1}{\cos^{3} (x) \cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 28

Multipliziere $\cos^{3} (x)$ mit $\cos (x)$ durch Addieren der Exponenten.

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Schritt 28.1

Mutltipliziere $\cos^{3} (x)$ mit $\cos (x)$ .

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Schritt 28.1.1

Potenziere $\cos (x)$ mit $1$ .

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x) \cdot 1}{\cos^{3} (x) \cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 28.1.2

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x) \cdot 1}{{\cos (x)}^{3 + 1}} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 28.2

Addiere $3$ und $1$ .

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x) \cdot 1}{\cos^{4} (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 29

Mutltipliziere $\sin^{4} (x)$ mit $1$ .

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x)}{\cos^{4} (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 30

Wandle von $\frac{\sin^{4} (x)}{\cos^{4} (x)}$ nach $\tan^{4} (x)$ um.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 31

Multipliziere den Zähler mit dem Kehrwert des Nenners.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \frac{1}{\cos^{3} (x)} \cdot \frac{1}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 32

Wandle von $\frac{1}{\cos (x)}$ nach $\sec (x)$ um.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \frac{1}{\cos^{3} (x)} \cdot \sec (x) = \frac{0}{\cos (x)}$

Schritt 33

Kombiniere $\sec (x)$ und $\frac{1}{\cos^{3} (x)}$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \frac{\sec (x)}{\cos^{3} (x)} = \frac{0}{\cos (x)}$

Schritt 34

Faktorisiere $\cos (x)$ aus $\cos^{3} (x)$ heraus.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \frac{\sec (x)}{\cos (x) \cos^{2} (x)} = \frac{0}{\cos (x)}$

Schritt 35

Separiere Brüche.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot \frac{\sec (x)}{\cos (x)}) = \frac{0}{\cos (x)}$

Schritt 36

Schreibe $\sec (x)$ mithilfe von Sinus und Kosinus um.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot \frac{\frac{1}{\cos (x)}}{\cos (x)}) = \frac{0}{\cos (x)}$

Schritt 37

Schreibe $\frac{\frac{1}{\cos (x)}}{\cos (x)}$ als ein Produkt um.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot (\frac{1}{\cos (x)} \cdot \frac{1}{\cos (x)})) = \frac{0}{\cos (x)}$

Schritt 38

Vereinfache.

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Schritt 38.1

Wandle von $\frac{1}{\cos (x)}$ nach $\sec (x)$ um.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot (\sec (x) (\frac{1}{\cos (x)}))) = \frac{0}{\cos (x)}$

Schritt 38.2

Wandle von $\frac{1}{\cos (x)}$ nach $\sec (x)$ um.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot (\sec (x) \sec (x))) = \frac{0}{\cos (x)}$

Schritt 38.3

Potenziere $\sec (x)$ mit $1$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot (\sec (x) \sec (x))) = \frac{0}{\cos (x)}$

Schritt 38.4

Potenziere $\sec (x)$ mit $1$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot (\sec (x) \sec (x))) = \frac{0}{\cos (x)}$

Schritt 38.5

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot {\sec (x)}^{1 + 1}) = \frac{0}{\cos (x)}$

Schritt 38.6

Addiere $1$ und $1$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot \sec^{2} (x)) = \frac{0}{\cos (x)}$

Schritt 39

Schreibe $1$ als $1^{2}$ um.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1^{2}}{\cos^{2} (x)} \cdot \sec^{2} (x)) = \frac{0}{\cos (x)}$

Schritt 40

Schreibe $\frac{1^{2}}{\cos^{2} (x)}$ als ${(\frac{1}{\cos (x)})}^{2}$ um.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - ({(\frac{1}{\cos (x)})}^{2} \sec^{2} (x)) = \frac{0}{\cos (x)}$

Schritt 41

Wandle von $\frac{1}{\cos (x)}$ nach $\sec (x)$ um.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\sec^{2} (x) \sec^{2} (x)) = \frac{0}{\cos (x)}$

Schritt 42

Multipliziere $\sec^{2} (x)$ mit $\sec^{2} (x)$ durch Addieren der Exponenten.

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Schritt 42.1

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - {\sec (x)}^{2 + 2} = \frac{0}{\cos (x)}$

Schritt 42.2

Addiere $2$ und $2$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = \frac{0}{\cos (x)}$

Schritt 43

Separiere Brüche.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = \frac{0}{1} \cdot \frac{1}{\cos (x)}$

Schritt 44

Wandle von $\frac{1}{\cos (x)}$ nach $\sec (x)$ um.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = \frac{0}{1} \cdot \sec (x)$

Schritt 45

Dividiere $0$ durch $1$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = 0 \sec (x)$

Schritt 46

Mutltipliziere $0$ mit $\sec (x)$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = 0$

Schritt 47

Ersetze die $2 \tan^{2} (x)$ durch $2 (\sec^{2} (x) - 1)$ basierend auf der $\tan^{2} (x) + 1 = \sec^{2} (x)$ -Identitätsgleichung.

$1 + 2 (\sec^{2} (x) - 1) + \tan^{4} (x) - \sec^{4} (x) = 0$

Schritt 48

Vereinfache jeden Term.

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Schritt 48.1

Wende das Distributivgesetz an.

$1 + 2 \sec^{2} (x) + 2 \cdot - 1 + \tan^{4} (x) - \sec^{4} (x) = 0$

Schritt 48.2

Mutltipliziere $2$ mit $- 1$ .

$1 + 2 \sec^{2} (x) - 2 + \tan^{4} (x) - \sec^{4} (x) = 0$

Schritt 49

Subtrahiere $2$ von $1$ .

$2 \sec^{2} (x) - 1 + \tan^{4} (x) - \sec^{4} (x) = 0$

Schritt 50

Stelle das Polynom um.

$\tan^{4} (x) - \sec^{4} (x) + 2 \sec^{2} (x) - 1 = 0$

Schritt 51

Multipliziere beide Seiten mit $\cos (x)$ .

$(1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 52

Vereinfache.

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Schritt 52.1

Vereinfache die linke Seite.

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Schritt 52.1.1

Vereinfache $(1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x)) \cos (x)$ .

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Schritt 52.1.1.1

Vereinfache jeden Term.

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Schritt 52.1.1.1.1

Schreibe $\tan (x)$ mithilfe von Sinus und Kosinus um.

$(1 + 2 {(\frac{\sin (x)}{\cos (x)})}^{2} + \tan^{4} (x) - \sec^{4} (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 52.1.1.1.2

Wende die Produktregel auf $\frac{\sin (x)}{\cos (x)}$ an.

$(1 + 2 \frac{\sin^{2} (x)}{\cos^{2} (x)} + \tan^{4} (x) - \sec^{4} (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 52.1.1.1.3

Kombiniere $2$ und $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ .

$(1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + \tan^{4} (x) - \sec^{4} (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 52.1.1.1.4

Schreibe $\tan (x)$ mithilfe von Sinus und Kosinus um.

$(1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + {(\frac{\sin (x)}{\cos (x)})}^{4} - \sec^{4} (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 52.1.1.1.5

Wende die Produktregel auf $\frac{\sin (x)}{\cos (x)}$ an.

$(1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + \frac{\sin^{4} (x)}{\cos^{4} (x)} - \sec^{4} (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 52.1.1.1.6

Schreibe $\sec (x)$ mithilfe von Sinus und Kosinus um.

$(1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + \frac{\sin^{4} (x)}{\cos^{4} (x)} - {(\frac{1}{\cos (x)})}^{4}) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 52.1.1.1.7

Wende die Produktregel auf $\frac{1}{\cos (x)}$ an.

$(1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + \frac{\sin^{4} (x)}{\cos^{4} (x)} - \frac{1^{4}}{\cos^{4} (x)}) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 52.1.1.1.8

Eins zu einer beliebigen Potenz erhoben ergibt eins.

$(1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + \frac{\sin^{4} (x)}{\cos^{4} (x)} - \frac{1}{\cos^{4} (x)}) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 52.1.1.2

Wende das Distributivgesetz an.

$1 \cos (x) + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} \cos (x) + \frac{\sin^{4} (x)}{\cos^{4} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 52.1.1.3

Vereinfache.

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Schritt 52.1.1.3.1

Mutltipliziere $\cos (x)$ mit $1$ .

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} \cos (x) + \frac{\sin^{4} (x)}{\cos^{4} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 52.1.1.3.2

Kürze den gemeinsamen Faktor von $\cos (x)$ .

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Schritt 52.1.1.3.2.1

Faktorisiere $\cos (x)$ aus $\cos^{2} (x)$ heraus.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x) \cos (x)} \cos (x) + \frac{\sin^{4} (x)}{\cos^{4} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 52.1.1.3.2.2

Kürze den gemeinsamen Faktor.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x) \cos (x)} \cos (x) + \frac{\sin^{4} (x)}{\cos^{4} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 52.1.1.3.2.3

Forme den Ausdruck um.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{4} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 52.1.1.3.3

Kürze den gemeinsamen Faktor von $\cos (x)$ .

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Schritt 52.1.1.3.3.1

Faktorisiere $\cos (x)$ aus $\cos^{4} (x)$ heraus.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos (x) \cos^{3} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 52.1.1.3.3.2

Kürze den gemeinsamen Faktor.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos (x) \cos^{3} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 52.1.1.3.3.3

Forme den Ausdruck um.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 52.1.1.3.4

Kürze den gemeinsamen Faktor von $\cos (x)$ .

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Schritt 52.1.1.3.4.1

Bringe das führende Minuszeichen in $- \frac{1}{\cos^{4} (x)}$ in den Zähler.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} + \frac{- 1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 52.1.1.3.4.2

Faktorisiere $\cos (x)$ aus $\cos^{4} (x)$ heraus.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} + \frac{- 1}{\cos (x) \cos^{3} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 52.1.1.3.4.3

Kürze den gemeinsamen Faktor.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} + \frac{- 1}{\cos (x) \cos^{3} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 52.1.1.3.4.4

Forme den Ausdruck um.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} + \frac{- 1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Schritt 52.1.1.4

Ziehe das Minuszeichen vor den Bruch.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Schritt 52.1.1.5

Vereinfache jeden Term.

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Schritt 52.1.1.5.1

Faktorisiere $\sin (x)$ aus $\sin^{2} (x)$ heraus.

$\cos (x) + \frac{2 (\sin (x) \sin (x))}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Schritt 52.1.1.5.2

Separiere Brüche.

$\cos (x) + \frac{2 (\sin (x))}{1} \cdot \frac{\sin (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Schritt 52.1.1.5.3

Wandle von $\frac{\sin (x)}{\cos (x)}$ nach $\tan (x)$ um.

$\cos (x) + \frac{2 (\sin (x))}{1} \tan (x) + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Schritt 52.1.1.5.4

Dividiere $2 (\sin (x))$ durch $1$ .

$\cos (x) + 2 (\sin (x)) \tan (x) + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Schritt 52.1.1.5.5

Faktorisiere $\sin^{3} (x)$ aus $\sin^{4} (x)$ heraus.

$\cos (x) + 2 \sin (x) \tan (x) + \frac{\sin^{3} (x) \sin (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Schritt 52.1.1.5.6

Multipliziere mit $1$ .

$\cos (x) + 2 \sin (x) \tan (x) + \frac{\sin^{3} (x) \sin (x)}{\cos^{3} (x) \cdot 1} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Schritt 52.1.1.5.7

Separiere Brüche.

$\cos (x) + 2 \sin (x) \tan (x) + \frac{\sin (x)}{1} \cdot \frac{\sin^{3} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Schritt 52.1.1.5.8

Wandle von $\frac{\sin^{3} (x)}{\cos^{3} (x)}$ nach $\tan^{3} (x)$ um.

$\cos (x) + 2 \sin (x) \tan (x) + \frac{\sin (x)}{1} \tan^{3} (x) - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Schritt 52.1.1.5.9

Dividiere $\sin (x)$ durch $1$ .

$\cos (x) + 2 \sin (x) \tan (x) + \sin (x) \tan^{3} (x) - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Schritt 52.2

Vereinfache die rechte Seite.

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Schritt 52.2.1

Kürze den gemeinsamen Faktor von $\cos (x)$ .

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Schritt 52.2.1.1

Kürze den gemeinsamen Faktor.

$\cos (x) + 2 \sin (x) \tan (x) + \sin (x) \tan^{3} (x) - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Schritt 52.2.1.2

Forme den Ausdruck um.

$\cos (x) + 2 \sin (x) \tan (x) + \sin (x) \tan^{3} (x) - \frac{1}{\cos^{3} (x)} = 0$

Schritt 53

Löse nach $x$ auf.

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Schritt 53.1

Teile jeden Term in der Gleichung durch $\cos (x)$ .

$\frac{\cos (x)}{\cos (x)} + \frac{2 \sin (x) \tan (x)}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.2

Kürze den gemeinsamen Faktor von $\cos (x)$ .

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Schritt 53.2.1

Kürze den gemeinsamen Faktor.

$\frac{\cos (x)}{\cos (x)} + \frac{2 \sin (x) \tan (x)}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.2.2

Forme den Ausdruck um.

$1 + \frac{2 \sin (x) \tan (x)}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.3

Vereinfache den Zähler.

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Schritt 53.3.1

Schreibe $\tan (x)$ mithilfe von Sinus und Kosinus um.

$1 + \frac{2 \sin (x) (\frac{\sin (x)}{\cos (x)})}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.3.2

Kombiniere Exponenten.

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Schritt 53.3.2.1

Kombiniere $\frac{\sin (x)}{\cos (x)}$ und $2$ .

$1 + \frac{\frac{\sin (x) \cdot 2}{\cos (x)} \cdot \sin (x)}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.3.2.2

Kombiniere $\frac{\sin (x) \cdot 2}{\cos (x)}$ und $\sin (x)$ .

$1 + \frac{\frac{\sin (x) \cdot (2 \sin (x))}{\cos (x)}}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.3.2.3

Potenziere $\sin (x)$ mit $1$ .

$1 + \frac{\frac{2 (\sin (x) \sin (x))}{\cos (x)}}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.3.2.4

Potenziere $\sin (x)$ mit $1$ .

$1 + \frac{\frac{2 (\sin (x) \sin (x))}{\cos (x)}}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.3.2.5

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$1 + \frac{\frac{2 {\sin (x)}^{1 + 1}}{\cos (x)}}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.3.2.6

Addiere $1$ und $1$ .

$1 + \frac{\frac{2 \sin^{2} (x)}{\cos (x)}}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.4

Multipliziere den Zähler mit dem Kehrwert des Nenners.

$1 + \frac{2 \sin^{2} (x)}{\cos (x)} \cdot \frac{1}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.5

Kombiniere Brüche.

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Schritt 53.5.1

Kombinieren.

$1 + \frac{2 \sin^{2} (x) \cdot 1}{\cos (x) \cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.5.2

Mutltipliziere $2$ mit $1$ .

$1 + \frac{2 \sin^{2} (x)}{\cos (x) \cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.6

Vereinfache den Nenner.

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Schritt 53.6.1

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$1 + \frac{2 \sin^{2} (x)}{{\cos (x)}^{1 + 1}} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.6.2

Addiere $1$ und $1$ .

$1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.7

Multipliziere mit $1$ .

$1 + \frac{2 (\sin^{2} (x) \cdot 1)}{\cos^{2} (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.8

Multipliziere mit $1$ .

$1 + \frac{2 (\sin^{2} (x) \cdot 1)}{\cos^{2} (x) \cdot 1} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.9

Separiere Brüche.

$1 + \frac{2 \cdot (1)}{1} \cdot \frac{\sin^{2} (x)}{\cos^{2} (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.10

Wandle von $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ nach $\tan^{2} (x)$ um.

$1 + \frac{2 \cdot (1)}{1} \cdot \tan^{2} (x) + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.11

Mutltipliziere $2$ mit $1$ .

$1 + \frac{2}{1} \cdot \tan^{2} (x) + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.12

Dividiere $2$ durch $1$ .

$1 + 2 \tan^{2} (x) + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.13

Vereinfache den Zähler.

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Schritt 53.13.1

Schreibe $\tan (x)$ mithilfe von Sinus und Kosinus um.

$1 + 2 \tan^{2} (x) + \frac{\sin (x) {(\frac{\sin (x)}{\cos (x)})}^{3}}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.13.2

Wende die Produktregel auf $\frac{\sin (x)}{\cos (x)}$ an.

$1 + 2 \tan^{2} (x) + \frac{\sin (x) (\frac{\sin^{3} (x)}{\cos^{3} (x)})}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.14

Kombiniere $\sin (x)$ und $\frac{\sin^{3} (x)}{\cos^{3} (x)}$ .

$1 + 2 \tan^{2} (x) + \frac{\frac{\sin (x) \sin^{3} (x)}{\cos^{3} (x)}}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.15

Multipliziere $\sin (x)$ mit $\sin^{3} (x)$ durch Addieren der Exponenten.

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Schritt 53.15.1

Mutltipliziere $\sin (x)$ mit $\sin^{3} (x)$ .

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Schritt 53.15.1.1

Potenziere $\sin (x)$ mit $1$ .

$1 + 2 \tan^{2} (x) + \frac{\frac{\sin (x) \sin^{3} (x)}{\cos^{3} (x)}}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.15.1.2

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$1 + 2 \tan^{2} (x) + \frac{\frac{{\sin (x)}^{1 + 3}}{\cos^{3} (x)}}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.15.2

Addiere $1$ und $3$ .

$1 + 2 \tan^{2} (x) + \frac{\frac{\sin^{4} (x)}{\cos^{3} (x)}}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.16

Multipliziere den Zähler mit dem Kehrwert des Nenners.

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x)}{\cos^{3} (x)} \cdot \frac{1}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.17

Kombinieren.

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x) \cdot 1}{\cos^{3} (x) \cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.18

Multipliziere $\cos^{3} (x)$ mit $\cos (x)$ durch Addieren der Exponenten.

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Schritt 53.18.1

Mutltipliziere $\cos^{3} (x)$ mit $\cos (x)$ .

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Schritt 53.18.1.1

Potenziere $\cos (x)$ mit $1$ .

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x) \cdot 1}{\cos^{3} (x) \cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.18.1.2

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x) \cdot 1}{{\cos (x)}^{3 + 1}} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.18.2

Addiere $3$ und $1$ .

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x) \cdot 1}{\cos^{4} (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.19

Mutltipliziere $\sin^{4} (x)$ mit $1$ .

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x)}{\cos^{4} (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.20

Wandle von $\frac{\sin^{4} (x)}{\cos^{4} (x)}$ nach $\tan^{4} (x)$ um.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.21

Multipliziere den Zähler mit dem Kehrwert des Nenners.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \frac{1}{\cos^{3} (x)} \cdot \frac{1}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.22

Wandle von $\frac{1}{\cos (x)}$ nach $\sec (x)$ um.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \frac{1}{\cos^{3} (x)} \cdot \sec (x) = \frac{0}{\cos (x)}$

Schritt 53.23

Kombiniere $\sec (x)$ und $\frac{1}{\cos^{3} (x)}$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \frac{\sec (x)}{\cos^{3} (x)} = \frac{0}{\cos (x)}$

Schritt 53.24

Faktorisiere $\cos (x)$ aus $\cos^{3} (x)$ heraus.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \frac{\sec (x)}{\cos (x) \cos^{2} (x)} = \frac{0}{\cos (x)}$

Schritt 53.25

Separiere Brüche.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot \frac{\sec (x)}{\cos (x)}) = \frac{0}{\cos (x)}$

Schritt 53.26

Schreibe $\sec (x)$ mithilfe von Sinus und Kosinus um.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot \frac{\frac{1}{\cos (x)}}{\cos (x)}) = \frac{0}{\cos (x)}$

Schritt 53.27

Schreibe $\frac{\frac{1}{\cos (x)}}{\cos (x)}$ als ein Produkt um.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot (\frac{1}{\cos (x)} \cdot \frac{1}{\cos (x)})) = \frac{0}{\cos (x)}$

Schritt 53.28

Vereinfache.

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Schritt 53.28.1

Wandle von $\frac{1}{\cos (x)}$ nach $\sec (x)$ um.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot (\sec (x) (\frac{1}{\cos (x)}))) = \frac{0}{\cos (x)}$

Schritt 53.28.2

Wandle von $\frac{1}{\cos (x)}$ nach $\sec (x)$ um.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot (\sec (x) \sec (x))) = \frac{0}{\cos (x)}$

Schritt 53.28.3

Potenziere $\sec (x)$ mit $1$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot (\sec (x) \sec (x))) = \frac{0}{\cos (x)}$

Schritt 53.28.4

Potenziere $\sec (x)$ mit $1$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot (\sec (x) \sec (x))) = \frac{0}{\cos (x)}$

Schritt 53.28.5

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot {\sec (x)}^{1 + 1}) = \frac{0}{\cos (x)}$

Schritt 53.28.6

Addiere $1$ und $1$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot \sec^{2} (x)) = \frac{0}{\cos (x)}$

Schritt 53.29

Schreibe $1$ als $1^{2}$ um.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1^{2}}{\cos^{2} (x)} \cdot \sec^{2} (x)) = \frac{0}{\cos (x)}$

Schritt 53.30

Schreibe $\frac{1^{2}}{\cos^{2} (x)}$ als ${(\frac{1}{\cos (x)})}^{2}$ um.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - ({(\frac{1}{\cos (x)})}^{2} \sec^{2} (x)) = \frac{0}{\cos (x)}$

Schritt 53.31

Wandle von $\frac{1}{\cos (x)}$ nach $\sec (x)$ um.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\sec^{2} (x) \sec^{2} (x)) = \frac{0}{\cos (x)}$

Schritt 53.32

Multipliziere $\sec^{2} (x)$ mit $\sec^{2} (x)$ durch Addieren der Exponenten.

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Schritt 53.32.1

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - {\sec (x)}^{2 + 2} = \frac{0}{\cos (x)}$

Schritt 53.32.2

Addiere $2$ und $2$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = \frac{0}{\cos (x)}$

Schritt 53.33

Separiere Brüche.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = \frac{0}{1} \cdot \frac{1}{\cos (x)}$

Schritt 53.34

Wandle von $\frac{1}{\cos (x)}$ nach $\sec (x)$ um.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = \frac{0}{1} \cdot \sec (x)$

Schritt 53.35

Dividiere $0$ durch $1$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = 0 \sec (x)$

Schritt 53.36

Mutltipliziere $0$ mit $\sec (x)$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = 0$

Schritt 53.37

Ersetze die $2 \tan^{2} (x)$ durch $2 (\sec^{2} (x) - 1)$ basierend auf der $\tan^{2} (x) + 1 = \sec^{2} (x)$ -Identitätsgleichung.

$1 + 2 (\sec^{2} (x) - 1) + \tan^{4} (x) - \sec^{4} (x) = 0$

Schritt 53.38

Vereinfache jeden Term.

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Schritt 53.38.1

Wende das Distributivgesetz an.

$1 + 2 \sec^{2} (x) + 2 \cdot - 1 + \tan^{4} (x) - \sec^{4} (x) = 0$

Schritt 53.38.2

Mutltipliziere $2$ mit $- 1$ .

$1 + 2 \sec^{2} (x) - 2 + \tan^{4} (x) - \sec^{4} (x) = 0$

Schritt 53.39

Subtrahiere $2$ von $1$ .

$2 \sec^{2} (x) - 1 + \tan^{4} (x) - \sec^{4} (x) = 0$

Schritt 53.40

Stelle das Polynom um.

$\tan^{4} (x) - \sec^{4} (x) + 2 \sec^{2} (x) - 1 = 0$

Schritt 53.41

Multipliziere beide Seiten mit $\cos (x)$ .

$(1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.42

Vereinfache.

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Schritt 53.42.1

Vereinfache die linke Seite.

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Schritt 53.42.1.1

Vereinfache $(1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x)) \cos (x)$ .

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Schritt 53.42.1.1.1

Vereinfache jeden Term.

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Schritt 53.42.1.1.1.1

Schreibe $\tan (x)$ mithilfe von Sinus und Kosinus um.

$(1 + 2 {(\frac{\sin (x)}{\cos (x)})}^{2} + \tan^{4} (x) - \sec^{4} (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.42.1.1.1.2

Wende die Produktregel auf $\frac{\sin (x)}{\cos (x)}$ an.

$(1 + 2 \frac{\sin^{2} (x)}{\cos^{2} (x)} + \tan^{4} (x) - \sec^{4} (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.42.1.1.1.3

Kombiniere $2$ und $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ .

$(1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + \tan^{4} (x) - \sec^{4} (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.42.1.1.1.4

Schreibe $\tan (x)$ mithilfe von Sinus und Kosinus um.

$(1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + {(\frac{\sin (x)}{\cos (x)})}^{4} - \sec^{4} (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.42.1.1.1.5

Wende die Produktregel auf $\frac{\sin (x)}{\cos (x)}$ an.

$(1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + \frac{\sin^{4} (x)}{\cos^{4} (x)} - \sec^{4} (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.42.1.1.1.6

Schreibe $\sec (x)$ mithilfe von Sinus und Kosinus um.

$(1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + \frac{\sin^{4} (x)}{\cos^{4} (x)} - {(\frac{1}{\cos (x)})}^{4}) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.42.1.1.1.7

Wende die Produktregel auf $\frac{1}{\cos (x)}$ an.

$(1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + \frac{\sin^{4} (x)}{\cos^{4} (x)} - \frac{1^{4}}{\cos^{4} (x)}) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.42.1.1.1.8

Eins zu einer beliebigen Potenz erhoben ergibt eins.

$(1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + \frac{\sin^{4} (x)}{\cos^{4} (x)} - \frac{1}{\cos^{4} (x)}) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.42.1.1.2

Wende das Distributivgesetz an.

$1 \cos (x) + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} \cos (x) + \frac{\sin^{4} (x)}{\cos^{4} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.42.1.1.3

Vereinfache.

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Schritt 53.42.1.1.3.1

Mutltipliziere $\cos (x)$ mit $1$ .

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} \cos (x) + \frac{\sin^{4} (x)}{\cos^{4} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.42.1.1.3.2

Kürze den gemeinsamen Faktor von $\cos (x)$ .

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Schritt 53.42.1.1.3.2.1

Faktorisiere $\cos (x)$ aus $\cos^{2} (x)$ heraus.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x) \cos (x)} \cos (x) + \frac{\sin^{4} (x)}{\cos^{4} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.42.1.1.3.2.2

Kürze den gemeinsamen Faktor.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x) \cos (x)} \cos (x) + \frac{\sin^{4} (x)}{\cos^{4} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.42.1.1.3.2.3

Forme den Ausdruck um.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{4} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.42.1.1.3.3

Kürze den gemeinsamen Faktor von $\cos (x)$ .

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Schritt 53.42.1.1.3.3.1

Faktorisiere $\cos (x)$ aus $\cos^{4} (x)$ heraus.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos (x) \cos^{3} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.42.1.1.3.3.2

Kürze den gemeinsamen Faktor.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos (x) \cos^{3} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.42.1.1.3.3.3

Forme den Ausdruck um.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.42.1.1.3.4

Kürze den gemeinsamen Faktor von $\cos (x)$ .

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Schritt 53.42.1.1.3.4.1

Bringe das führende Minuszeichen in $- \frac{1}{\cos^{4} (x)}$ in den Zähler.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} + \frac{- 1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.42.1.1.3.4.2

Faktorisiere $\cos (x)$ aus $\cos^{4} (x)$ heraus.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} + \frac{- 1}{\cos (x) \cos^{3} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.42.1.1.3.4.3

Kürze den gemeinsamen Faktor.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} + \frac{- 1}{\cos (x) \cos^{3} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.42.1.1.3.4.4

Forme den Ausdruck um.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} + \frac{- 1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.42.1.1.4

Ziehe das Minuszeichen vor den Bruch.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.42.1.1.5

Vereinfache jeden Term.

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Schritt 53.42.1.1.5.1

Faktorisiere $\sin (x)$ aus $\sin^{2} (x)$ heraus.

$\cos (x) + \frac{2 (\sin (x) \sin (x))}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.42.1.1.5.2

Separiere Brüche.

$\cos (x) + \frac{2 (\sin (x))}{1} \cdot \frac{\sin (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.42.1.1.5.3

Wandle von $\frac{\sin (x)}{\cos (x)}$ nach $\tan (x)$ um.

$\cos (x) + \frac{2 (\sin (x))}{1} \tan (x) + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.42.1.1.5.4

Dividiere $2 (\sin (x))$ durch $1$ .

$\cos (x) + 2 (\sin (x)) \tan (x) + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.42.1.1.5.5

Faktorisiere $\sin^{3} (x)$ aus $\sin^{4} (x)$ heraus.

$\cos (x) + 2 \sin (x) \tan (x) + \frac{\sin^{3} (x) \sin (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.42.1.1.5.6

Multipliziere mit $1$ .

$\cos (x) + 2 \sin (x) \tan (x) + \frac{\sin^{3} (x) \sin (x)}{\cos^{3} (x) \cdot 1} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.42.1.1.5.7

Separiere Brüche.

$\cos (x) + 2 \sin (x) \tan (x) + \frac{\sin (x)}{1} \cdot \frac{\sin^{3} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.42.1.1.5.8

Wandle von $\frac{\sin^{3} (x)}{\cos^{3} (x)}$ nach $\tan^{3} (x)$ um.

$\cos (x) + 2 \sin (x) \tan (x) + \frac{\sin (x)}{1} \tan^{3} (x) - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.42.1.1.5.9

Dividiere $\sin (x)$ durch $1$ .

$\cos (x) + 2 \sin (x) \tan (x) + \sin (x) \tan^{3} (x) - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.42.2

Vereinfache die rechte Seite.

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Schritt 53.42.2.1

Kürze den gemeinsamen Faktor von $\cos (x)$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 53.42.2.1.1

Kürze den gemeinsamen Faktor.

$\cos (x) + 2 \sin (x) \tan (x) + \sin (x) \tan^{3} (x) - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.42.2.1.2

Forme den Ausdruck um.

$\cos (x) + 2 \sin (x) \tan (x) + \sin (x) \tan^{3} (x) - \frac{1}{\cos^{3} (x)} = 0$

Schritt 53.43

Löse nach $x$ auf.

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Schritt 53.43.1

Teile jeden Term in der Gleichung durch $\cos (x)$ .

$\frac{\cos (x)}{\cos (x)} + \frac{2 \sin (x) \tan (x)}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.2

Kürze den gemeinsamen Faktor von $\cos (x)$ .

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Schritt 53.43.2.1

Kürze den gemeinsamen Faktor.

$\frac{\cos (x)}{\cos (x)} + \frac{2 \sin (x) \tan (x)}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.2.2

Forme den Ausdruck um.

$1 + \frac{2 \sin (x) \tan (x)}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.3

Vereinfache den Zähler.

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Schritt 53.43.3.1

Schreibe $\tan (x)$ mithilfe von Sinus und Kosinus um.

$1 + \frac{2 \sin (x) (\frac{\sin (x)}{\cos (x)})}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.3.2

Kombiniere Exponenten.

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Schritt 53.43.3.2.1

Kombiniere $\frac{\sin (x)}{\cos (x)}$ und $2$ .

$1 + \frac{\frac{\sin (x) \cdot 2}{\cos (x)} \cdot \sin (x)}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.3.2.2

Kombiniere $\frac{\sin (x) \cdot 2}{\cos (x)}$ und $\sin (x)$ .

$1 + \frac{\frac{\sin (x) \cdot (2 \sin (x))}{\cos (x)}}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.3.2.3

Potenziere $\sin (x)$ mit $1$ .

$1 + \frac{\frac{2 (\sin (x) \sin (x))}{\cos (x)}}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.3.2.4

Potenziere $\sin (x)$ mit $1$ .

$1 + \frac{\frac{2 (\sin (x) \sin (x))}{\cos (x)}}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.3.2.5

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$1 + \frac{\frac{2 {\sin (x)}^{1 + 1}}{\cos (x)}}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.3.2.6

Addiere $1$ und $1$ .

$1 + \frac{\frac{2 \sin^{2} (x)}{\cos (x)}}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.4

Multipliziere den Zähler mit dem Kehrwert des Nenners.

$1 + \frac{2 \sin^{2} (x)}{\cos (x)} \cdot \frac{1}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.5

Kombiniere Brüche.

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Schritt 53.43.5.1

Kombinieren.

$1 + \frac{2 \sin^{2} (x) \cdot 1}{\cos (x) \cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.5.2

Mutltipliziere $2$ mit $1$ .

$1 + \frac{2 \sin^{2} (x)}{\cos (x) \cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.6

Vereinfache den Nenner.

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Schritt 53.43.6.1

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$1 + \frac{2 \sin^{2} (x)}{{\cos (x)}^{1 + 1}} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.6.2

Addiere $1$ und $1$ .

$1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.7

Multipliziere mit $1$ .

$1 + \frac{2 (\sin^{2} (x) \cdot 1)}{\cos^{2} (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.8

Multipliziere mit $1$ .

$1 + \frac{2 (\sin^{2} (x) \cdot 1)}{\cos^{2} (x) \cdot 1} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.9

Separiere Brüche.

$1 + \frac{2 \cdot (1)}{1} \cdot \frac{\sin^{2} (x)}{\cos^{2} (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.10

Wandle von $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ nach $\tan^{2} (x)$ um.

$1 + \frac{2 \cdot (1)}{1} \cdot \tan^{2} (x) + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.11

Mutltipliziere $2$ mit $1$ .

$1 + \frac{2}{1} \cdot \tan^{2} (x) + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.12

Dividiere $2$ durch $1$ .

$1 + 2 \tan^{2} (x) + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.13

Vereinfache den Zähler.

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Schritt 53.43.13.1

Schreibe $\tan (x)$ mithilfe von Sinus und Kosinus um.

$1 + 2 \tan^{2} (x) + \frac{\sin (x) {(\frac{\sin (x)}{\cos (x)})}^{3}}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.13.2

Wende die Produktregel auf $\frac{\sin (x)}{\cos (x)}$ an.

$1 + 2 \tan^{2} (x) + \frac{\sin (x) (\frac{\sin^{3} (x)}{\cos^{3} (x)})}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.14

Kombiniere $\sin (x)$ und $\frac{\sin^{3} (x)}{\cos^{3} (x)}$ .

$1 + 2 \tan^{2} (x) + \frac{\frac{\sin (x) \sin^{3} (x)}{\cos^{3} (x)}}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.15

Multipliziere $\sin (x)$ mit $\sin^{3} (x)$ durch Addieren der Exponenten.

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Schritt 53.43.15.1

Mutltipliziere $\sin (x)$ mit $\sin^{3} (x)$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 53.43.15.1.1

Potenziere $\sin (x)$ mit $1$ .

$1 + 2 \tan^{2} (x) + \frac{\frac{\sin (x) \sin^{3} (x)}{\cos^{3} (x)}}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.15.1.2

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$1 + 2 \tan^{2} (x) + \frac{\frac{{\sin (x)}^{1 + 3}}{\cos^{3} (x)}}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.15.2

Addiere $1$ und $3$ .

$1 + 2 \tan^{2} (x) + \frac{\frac{\sin^{4} (x)}{\cos^{3} (x)}}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.16

Multipliziere den Zähler mit dem Kehrwert des Nenners.

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x)}{\cos^{3} (x)} \cdot \frac{1}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.17

Kombinieren.

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x) \cdot 1}{\cos^{3} (x) \cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.18

Multipliziere $\cos^{3} (x)$ mit $\cos (x)$ durch Addieren der Exponenten.

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Schritt 53.43.18.1

Mutltipliziere $\cos^{3} (x)$ mit $\cos (x)$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 53.43.18.1.1

Potenziere $\cos (x)$ mit $1$ .

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x) \cdot 1}{\cos^{3} (x) \cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.18.1.2

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x) \cdot 1}{{\cos (x)}^{3 + 1}} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.18.2

Addiere $3$ und $1$ .

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x) \cdot 1}{\cos^{4} (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.19

Mutltipliziere $\sin^{4} (x)$ mit $1$ .

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x)}{\cos^{4} (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.20

Wandle von $\frac{\sin^{4} (x)}{\cos^{4} (x)}$ nach $\tan^{4} (x)$ um.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.21

Multipliziere den Zähler mit dem Kehrwert des Nenners.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \frac{1}{\cos^{3} (x)} \cdot \frac{1}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.22

Wandle von $\frac{1}{\cos (x)}$ nach $\sec (x)$ um.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \frac{1}{\cos^{3} (x)} \cdot \sec (x) = \frac{0}{\cos (x)}$

Schritt 53.43.23

Kombiniere $\sec (x)$ und $\frac{1}{\cos^{3} (x)}$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \frac{\sec (x)}{\cos^{3} (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.24

Faktorisiere $\cos (x)$ aus $\cos^{3} (x)$ heraus.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \frac{\sec (x)}{\cos (x) \cos^{2} (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.25

Separiere Brüche.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot \frac{\sec (x)}{\cos (x)}) = \frac{0}{\cos (x)}$

Schritt 53.43.26

Schreibe $\sec (x)$ mithilfe von Sinus und Kosinus um.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot \frac{\frac{1}{\cos (x)}}{\cos (x)}) = \frac{0}{\cos (x)}$

Schritt 53.43.27

Schreibe $\frac{\frac{1}{\cos (x)}}{\cos (x)}$ als ein Produkt um.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot (\frac{1}{\cos (x)} \cdot \frac{1}{\cos (x)})) = \frac{0}{\cos (x)}$

Schritt 53.43.28

Vereinfache.

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Schritt 53.43.28.1

Wandle von $\frac{1}{\cos (x)}$ nach $\sec (x)$ um.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot (\sec (x) (\frac{1}{\cos (x)}))) = \frac{0}{\cos (x)}$

Schritt 53.43.28.2

Wandle von $\frac{1}{\cos (x)}$ nach $\sec (x)$ um.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot (\sec (x) \sec (x))) = \frac{0}{\cos (x)}$

Schritt 53.43.28.3

Potenziere $\sec (x)$ mit $1$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot (\sec (x) \sec (x))) = \frac{0}{\cos (x)}$

Schritt 53.43.28.4

Potenziere $\sec (x)$ mit $1$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot (\sec (x) \sec (x))) = \frac{0}{\cos (x)}$

Schritt 53.43.28.5

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot {\sec (x)}^{1 + 1}) = \frac{0}{\cos (x)}$

Schritt 53.43.28.6

Addiere $1$ und $1$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot \sec^{2} (x)) = \frac{0}{\cos (x)}$

Schritt 53.43.29

Schreibe $1$ als $1^{2}$ um.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1^{2}}{\cos^{2} (x)} \cdot \sec^{2} (x)) = \frac{0}{\cos (x)}$

Schritt 53.43.30

Schreibe $\frac{1^{2}}{\cos^{2} (x)}$ als ${(\frac{1}{\cos (x)})}^{2}$ um.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - ({(\frac{1}{\cos (x)})}^{2} \sec^{2} (x)) = \frac{0}{\cos (x)}$

Schritt 53.43.31

Wandle von $\frac{1}{\cos (x)}$ nach $\sec (x)$ um.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\sec^{2} (x) \sec^{2} (x)) = \frac{0}{\cos (x)}$

Schritt 53.43.32

Multipliziere $\sec^{2} (x)$ mit $\sec^{2} (x)$ durch Addieren der Exponenten.

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Schritt 53.43.32.1

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - {\sec (x)}^{2 + 2} = \frac{0}{\cos (x)}$

Schritt 53.43.32.2

Addiere $2$ und $2$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = \frac{0}{\cos (x)}$

Schritt 53.43.33

Separiere Brüche.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = \frac{0}{1} \cdot \frac{1}{\cos (x)}$

Schritt 53.43.34

Wandle von $\frac{1}{\cos (x)}$ nach $\sec (x)$ um.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = \frac{0}{1} \cdot \sec (x)$

Schritt 53.43.35

Dividiere $0$ durch $1$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = 0 \sec (x)$

Schritt 53.43.36

Mutltipliziere $0$ mit $\sec (x)$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = 0$

Schritt 53.43.37

Ersetze die $2 \tan^{2} (x)$ durch $2 (\sec^{2} (x) - 1)$ basierend auf der $\tan^{2} (x) + 1 = \sec^{2} (x)$ -Identitätsgleichung.

$1 + 2 (\sec^{2} (x) - 1) + \tan^{4} (x) - \sec^{4} (x) = 0$

Schritt 53.43.38

Vereinfache jeden Term.

Tippen, um mehr Schritte zu sehen ...

Schritt 53.43.38.1

Wende das Distributivgesetz an.

$1 + 2 \sec^{2} (x) + 2 \cdot - 1 + \tan^{4} (x) - \sec^{4} (x) = 0$

Schritt 53.43.38.2

Mutltipliziere $2$ mit $- 1$ .

$1 + 2 \sec^{2} (x) - 2 + \tan^{4} (x) - \sec^{4} (x) = 0$

Schritt 53.43.39

Subtrahiere $2$ von $1$ .

$2 \sec^{2} (x) - 1 + \tan^{4} (x) - \sec^{4} (x) = 0$

Schritt 53.43.40

Stelle das Polynom um.

$\tan^{4} (x) - \sec^{4} (x) + 2 \sec^{2} (x) - 1 = 0$

Schritt 53.43.41

Multipliziere beide Seiten mit $\cos (x)$ .

$(1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.42

Vereinfache.

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Schritt 53.43.42.1

Vereinfache die linke Seite.

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Schritt 53.43.42.1.1

Vereinfache $(1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x)) \cos (x)$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 53.43.42.1.1.1

Vereinfache jeden Term.

Tippen, um mehr Schritte zu sehen ...

Schritt 53.43.42.1.1.1.1

Schreibe $\tan (x)$ mithilfe von Sinus und Kosinus um.

$(1 + 2 {(\frac{\sin (x)}{\cos (x)})}^{2} + \tan^{4} (x) - \sec^{4} (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.42.1.1.1.2

Wende die Produktregel auf $\frac{\sin (x)}{\cos (x)}$ an.

$(1 + 2 \frac{\sin^{2} (x)}{\cos^{2} (x)} + \tan^{4} (x) - \sec^{4} (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.42.1.1.1.3

Kombiniere $2$ und $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ .

$(1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + \tan^{4} (x) - \sec^{4} (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.42.1.1.1.4

Schreibe $\tan (x)$ mithilfe von Sinus und Kosinus um.

$(1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + {(\frac{\sin (x)}{\cos (x)})}^{4} - \sec^{4} (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.42.1.1.1.5

Wende die Produktregel auf $\frac{\sin (x)}{\cos (x)}$ an.

$(1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + \frac{\sin^{4} (x)}{\cos^{4} (x)} - \sec^{4} (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.42.1.1.1.6

Schreibe $\sec (x)$ mithilfe von Sinus und Kosinus um.

$(1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + \frac{\sin^{4} (x)}{\cos^{4} (x)} - {(\frac{1}{\cos (x)})}^{4}) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.42.1.1.1.7

Wende die Produktregel auf $\frac{1}{\cos (x)}$ an.

$(1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + \frac{\sin^{4} (x)}{\cos^{4} (x)} - \frac{1^{4}}{\cos^{4} (x)}) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.42.1.1.1.8

Eins zu einer beliebigen Potenz erhoben ergibt eins.

$(1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + \frac{\sin^{4} (x)}{\cos^{4} (x)} - \frac{1}{\cos^{4} (x)}) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.42.1.1.2

Wende das Distributivgesetz an.

$1 \cos (x) + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} \cos (x) + \frac{\sin^{4} (x)}{\cos^{4} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.42.1.1.3

Vereinfache.

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Schritt 53.43.42.1.1.3.1

Mutltipliziere $\cos (x)$ mit $1$ .

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} \cos (x) + \frac{\sin^{4} (x)}{\cos^{4} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.42.1.1.3.2

Kürze den gemeinsamen Faktor von $\cos (x)$ .

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Schritt 53.43.42.1.1.3.2.1

Faktorisiere $\cos (x)$ aus $\cos^{2} (x)$ heraus.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x) \cos (x)} \cos (x) + \frac{\sin^{4} (x)}{\cos^{4} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.42.1.1.3.2.2

Kürze den gemeinsamen Faktor.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x) \cos (x)} \cos (x) + \frac{\sin^{4} (x)}{\cos^{4} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.42.1.1.3.2.3

Forme den Ausdruck um.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{4} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.42.1.1.3.3

Kürze den gemeinsamen Faktor von $\cos (x)$ .

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Schritt 53.43.42.1.1.3.3.1

Faktorisiere $\cos (x)$ aus $\cos^{4} (x)$ heraus.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos (x) \cos^{3} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.42.1.1.3.3.2

Kürze den gemeinsamen Faktor.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos (x) \cos^{3} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.42.1.1.3.3.3

Forme den Ausdruck um.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.42.1.1.3.4

Kürze den gemeinsamen Faktor von $\cos (x)$ .

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Schritt 53.43.42.1.1.3.4.1

Bringe das führende Minuszeichen in $- \frac{1}{\cos^{4} (x)}$ in den Zähler.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} + \frac{- 1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.42.1.1.3.4.2

Faktorisiere $\cos (x)$ aus $\cos^{4} (x)$ heraus.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} + \frac{- 1}{\cos (x) \cos^{3} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.42.1.1.3.4.3

Kürze den gemeinsamen Faktor.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} + \frac{- 1}{\cos (x) \cos^{3} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.42.1.1.3.4.4

Forme den Ausdruck um.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} + \frac{- 1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.42.1.1.4

Ziehe das Minuszeichen vor den Bruch.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.42.1.1.5

Vereinfache jeden Term.

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Schritt 53.43.42.1.1.5.1

Faktorisiere $\sin (x)$ aus $\sin^{2} (x)$ heraus.

$\cos (x) + \frac{2 (\sin (x) \sin (x))}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.42.1.1.5.2

Separiere Brüche.

$\cos (x) + \frac{2 (\sin (x))}{1} \cdot \frac{\sin (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.42.1.1.5.3

Wandle von $\frac{\sin (x)}{\cos (x)}$ nach $\tan (x)$ um.

$\cos (x) + \frac{2 (\sin (x))}{1} \tan (x) + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.42.1.1.5.4

Dividiere $2 (\sin (x))$ durch $1$ .

$\cos (x) + 2 (\sin (x)) \tan (x) + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.42.1.1.5.5

Faktorisiere $\sin^{3} (x)$ aus $\sin^{4} (x)$ heraus.

$\cos (x) + 2 \sin (x) \tan (x) + \frac{\sin^{3} (x) \sin (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.42.1.1.5.6

Multipliziere mit $1$ .

$\cos (x) + 2 \sin (x) \tan (x) + \frac{\sin^{3} (x) \sin (x)}{\cos^{3} (x) \cdot 1} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.42.1.1.5.7

Separiere Brüche.

$\cos (x) + 2 \sin (x) \tan (x) + \frac{\sin (x)}{1} \cdot \frac{\sin^{3} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.42.1.1.5.8

Wandle von $\frac{\sin^{3} (x)}{\cos^{3} (x)}$ nach $\tan^{3} (x)$ um.

$\cos (x) + 2 \sin (x) \tan (x) + \frac{\sin (x)}{1} \tan^{3} (x) - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.42.1.1.5.9

Dividiere $\sin (x)$ durch $1$ .

$\cos (x) + 2 \sin (x) \tan (x) + \sin (x) \tan^{3} (x) - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.42.2

Vereinfache die rechte Seite.

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Schritt 53.43.42.2.1

Kürze den gemeinsamen Faktor von $\cos (x)$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 53.43.42.2.1.1

Kürze den gemeinsamen Faktor.

$\cos (x) + 2 \sin (x) \tan (x) + \sin (x) \tan^{3} (x) - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.42.2.1.2

Forme den Ausdruck um.

$\cos (x) + 2 \sin (x) \tan (x) + \sin (x) \tan^{3} (x) - \frac{1}{\cos^{3} (x)} = 0$

Schritt 53.43.43

Löse nach $x$ auf.

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Schritt 53.43.43.1

Teile jeden Term in der Gleichung durch $\cos (x)$ .

$\frac{\cos (x)}{\cos (x)} + \frac{2 \sin (x) \tan (x)}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.2

Kürze den gemeinsamen Faktor von $\cos (x)$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 53.43.43.2.1

Kürze den gemeinsamen Faktor.

$\frac{\cos (x)}{\cos (x)} + \frac{2 \sin (x) \tan (x)}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.2.2

Forme den Ausdruck um.

$1 + \frac{2 \sin (x) \tan (x)}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.3

Vereinfache den Zähler.

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Schritt 53.43.43.3.1

Schreibe $\tan (x)$ mithilfe von Sinus und Kosinus um.

$1 + \frac{2 \sin (x) (\frac{\sin (x)}{\cos (x)})}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.3.2

Kombiniere Exponenten.

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Schritt 53.43.43.3.2.1

Kombiniere $\frac{\sin (x)}{\cos (x)}$ und $2$ .

$1 + \frac{\frac{\sin (x) \cdot 2}{\cos (x)} \cdot \sin (x)}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.3.2.2

Kombiniere $\frac{\sin (x) \cdot 2}{\cos (x)}$ und $\sin (x)$ .

$1 + \frac{\frac{\sin (x) \cdot (2 \sin (x))}{\cos (x)}}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.3.2.3

Potenziere $\sin (x)$ mit $1$ .

$1 + \frac{\frac{2 (\sin (x) \sin (x))}{\cos (x)}}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.3.2.4

Potenziere $\sin (x)$ mit $1$ .

$1 + \frac{\frac{2 (\sin (x) \sin (x))}{\cos (x)}}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.3.2.5

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$1 + \frac{\frac{2 {\sin (x)}^{1 + 1}}{\cos (x)}}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.3.2.6

Addiere $1$ und $1$ .

$1 + \frac{\frac{2 \sin^{2} (x)}{\cos (x)}}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.4

Multipliziere den Zähler mit dem Kehrwert des Nenners.

$1 + \frac{2 \sin^{2} (x)}{\cos (x)} \cdot \frac{1}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.5

Kombiniere Brüche.

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Schritt 53.43.43.5.1

Kombinieren.

$1 + \frac{2 \sin^{2} (x) \cdot 1}{\cos (x) \cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.5.2

Mutltipliziere $2$ mit $1$ .

$1 + \frac{2 \sin^{2} (x)}{\cos (x) \cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.6

Vereinfache den Nenner.

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Schritt 53.43.43.6.1

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$1 + \frac{2 \sin^{2} (x)}{{\cos (x)}^{1 + 1}} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.6.2

Addiere $1$ und $1$ .

$1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.7

Multipliziere mit $1$ .

$1 + \frac{2 (\sin^{2} (x) \cdot 1)}{\cos^{2} (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.8

Multipliziere mit $1$ .

$1 + \frac{2 (\sin^{2} (x) \cdot 1)}{\cos^{2} (x) \cdot 1} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.9

Separiere Brüche.

$1 + \frac{2 \cdot (1)}{1} \cdot \frac{\sin^{2} (x)}{\cos^{2} (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.10

Wandle von $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ nach $\tan^{2} (x)$ um.

$1 + \frac{2 \cdot (1)}{1} \cdot \tan^{2} (x) + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.11

Mutltipliziere $2$ mit $1$ .

$1 + \frac{2}{1} \cdot \tan^{2} (x) + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.12

Dividiere $2$ durch $1$ .

$1 + 2 \tan^{2} (x) + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.13

Vereinfache den Zähler.

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Schritt 53.43.43.13.1

Schreibe $\tan (x)$ mithilfe von Sinus und Kosinus um.

$1 + 2 \tan^{2} (x) + \frac{\sin (x) {(\frac{\sin (x)}{\cos (x)})}^{3}}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.13.2

Wende die Produktregel auf $\frac{\sin (x)}{\cos (x)}$ an.

$1 + 2 \tan^{2} (x) + \frac{\sin (x) (\frac{\sin^{3} (x)}{\cos^{3} (x)})}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.14

Kombiniere $\sin (x)$ und $\frac{\sin^{3} (x)}{\cos^{3} (x)}$ .

$1 + 2 \tan^{2} (x) + \frac{\frac{\sin (x) \sin^{3} (x)}{\cos^{3} (x)}}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.15

Multipliziere $\sin (x)$ mit $\sin^{3} (x)$ durch Addieren der Exponenten.

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Schritt 53.43.43.15.1

Mutltipliziere $\sin (x)$ mit $\sin^{3} (x)$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 53.43.43.15.1.1

Potenziere $\sin (x)$ mit $1$ .

$1 + 2 \tan^{2} (x) + \frac{\frac{\sin (x) \sin^{3} (x)}{\cos^{3} (x)}}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.15.1.2

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$1 + 2 \tan^{2} (x) + \frac{\frac{{\sin (x)}^{1 + 3}}{\cos^{3} (x)}}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.15.2

Addiere $1$ und $3$ .

$1 + 2 \tan^{2} (x) + \frac{\frac{\sin^{4} (x)}{\cos^{3} (x)}}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.16

Multipliziere den Zähler mit dem Kehrwert des Nenners.

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x)}{\cos^{3} (x)} \cdot \frac{1}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.17

Kombinieren.

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x) \cdot 1}{\cos^{3} (x) \cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.18

Multipliziere $\cos^{3} (x)$ mit $\cos (x)$ durch Addieren der Exponenten.

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Schritt 53.43.43.18.1

Mutltipliziere $\cos^{3} (x)$ mit $\cos (x)$ .

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Schritt 53.43.43.18.1.1

Potenziere $\cos (x)$ mit $1$ .

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x) \cdot 1}{\cos^{3} (x) \cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.18.1.2

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x) \cdot 1}{{\cos (x)}^{3 + 1}} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.18.2

Addiere $3$ und $1$ .

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x) \cdot 1}{\cos^{4} (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.19

Mutltipliziere $\sin^{4} (x)$ mit $1$ .

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x)}{\cos^{4} (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.20

Wandle von $\frac{\sin^{4} (x)}{\cos^{4} (x)}$ nach $\tan^{4} (x)$ um.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.21

Multipliziere den Zähler mit dem Kehrwert des Nenners.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \frac{1}{\cos^{3} (x)} \cdot \frac{1}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.22

Wandle von $\frac{1}{\cos (x)}$ nach $\sec (x)$ um.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \frac{1}{\cos^{3} (x)} \cdot \sec (x) = \frac{0}{\cos (x)}$

Schritt 53.43.43.23

Kombiniere $\sec (x)$ und $\frac{1}{\cos^{3} (x)}$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \frac{\sec (x)}{\cos^{3} (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.24

Faktorisiere $\cos (x)$ aus $\cos^{3} (x)$ heraus.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \frac{\sec (x)}{\cos (x) \cos^{2} (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.25

Separiere Brüche.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot \frac{\sec (x)}{\cos (x)}) = \frac{0}{\cos (x)}$

Schritt 53.43.43.26

Schreibe $\sec (x)$ mithilfe von Sinus und Kosinus um.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot \frac{\frac{1}{\cos (x)}}{\cos (x)}) = \frac{0}{\cos (x)}$

Schritt 53.43.43.27

Schreibe $\frac{\frac{1}{\cos (x)}}{\cos (x)}$ als ein Produkt um.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot (\frac{1}{\cos (x)} \cdot \frac{1}{\cos (x)})) = \frac{0}{\cos (x)}$

Schritt 53.43.43.28

Vereinfache.

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Schritt 53.43.43.28.1

Wandle von $\frac{1}{\cos (x)}$ nach $\sec (x)$ um.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot (\sec (x) (\frac{1}{\cos (x)}))) = \frac{0}{\cos (x)}$

Schritt 53.43.43.28.2

Wandle von $\frac{1}{\cos (x)}$ nach $\sec (x)$ um.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot (\sec (x) \sec (x))) = \frac{0}{\cos (x)}$

Schritt 53.43.43.28.3

Potenziere $\sec (x)$ mit $1$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot (\sec (x) \sec (x))) = \frac{0}{\cos (x)}$

Schritt 53.43.43.28.4

Potenziere $\sec (x)$ mit $1$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot (\sec (x) \sec (x))) = \frac{0}{\cos (x)}$

Schritt 53.43.43.28.5

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot {\sec (x)}^{1 + 1}) = \frac{0}{\cos (x)}$

Schritt 53.43.43.28.6

Addiere $1$ und $1$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot \sec^{2} (x)) = \frac{0}{\cos (x)}$

Schritt 53.43.43.29

Schreibe $1$ als $1^{2}$ um.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1^{2}}{\cos^{2} (x)} \cdot \sec^{2} (x)) = \frac{0}{\cos (x)}$

Schritt 53.43.43.30

Schreibe $\frac{1^{2}}{\cos^{2} (x)}$ als ${(\frac{1}{\cos (x)})}^{2}$ um.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - ({(\frac{1}{\cos (x)})}^{2} \sec^{2} (x)) = \frac{0}{\cos (x)}$

Schritt 53.43.43.31

Wandle von $\frac{1}{\cos (x)}$ nach $\sec (x)$ um.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\sec^{2} (x) \sec^{2} (x)) = \frac{0}{\cos (x)}$

Schritt 53.43.43.32

Multipliziere $\sec^{2} (x)$ mit $\sec^{2} (x)$ durch Addieren der Exponenten.

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Schritt 53.43.43.32.1

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - {\sec (x)}^{2 + 2} = \frac{0}{\cos (x)}$

Schritt 53.43.43.32.2

Addiere $2$ und $2$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = \frac{0}{\cos (x)}$

Schritt 53.43.43.33

Separiere Brüche.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = \frac{0}{1} \cdot \frac{1}{\cos (x)}$

Schritt 53.43.43.34

Wandle von $\frac{1}{\cos (x)}$ nach $\sec (x)$ um.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = \frac{0}{1} \cdot \sec (x)$

Schritt 53.43.43.35

Dividiere $0$ durch $1$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = 0 \sec (x)$

Schritt 53.43.43.36

Mutltipliziere $0$ mit $\sec (x)$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = 0$

Schritt 53.43.43.37

Ersetze die $2 \tan^{2} (x)$ durch $2 (\sec^{2} (x) - 1)$ basierend auf der $\tan^{2} (x) + 1 = \sec^{2} (x)$ -Identitätsgleichung.

$1 + 2 (\sec^{2} (x) - 1) + \tan^{4} (x) - \sec^{4} (x) = 0$

Schritt 53.43.43.38

Vereinfache jeden Term.

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Schritt 53.43.43.38.1

Wende das Distributivgesetz an.

$1 + 2 \sec^{2} (x) + 2 \cdot - 1 + \tan^{4} (x) - \sec^{4} (x) = 0$

Schritt 53.43.43.38.2

Mutltipliziere $2$ mit $- 1$ .

$1 + 2 \sec^{2} (x) - 2 + \tan^{4} (x) - \sec^{4} (x) = 0$

Schritt 53.43.43.39

Subtrahiere $2$ von $1$ .

$2 \sec^{2} (x) - 1 + \tan^{4} (x) - \sec^{4} (x) = 0$

Schritt 53.43.43.40

Stelle das Polynom um.

$\tan^{4} (x) - \sec^{4} (x) + 2 \sec^{2} (x) - 1 = 0$

Schritt 53.43.43.41

Multipliziere beide Seiten mit $\cos (x)$ .

$(1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.43.42

Vereinfache.

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Schritt 53.43.43.42.1

Vereinfache die linke Seite.

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Schritt 53.43.43.42.1.1

Vereinfache $(1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x)) \cos (x)$ .

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Schritt 53.43.43.42.1.1.1

Vereinfache jeden Term.

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Schritt 53.43.43.42.1.1.1.1

Schreibe $\tan (x)$ mithilfe von Sinus und Kosinus um.

$(1 + 2 {(\frac{\sin (x)}{\cos (x)})}^{2} + \tan^{4} (x) - \sec^{4} (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.43.42.1.1.1.2

Wende die Produktregel auf $\frac{\sin (x)}{\cos (x)}$ an.

$(1 + 2 \frac{\sin^{2} (x)}{\cos^{2} (x)} + \tan^{4} (x) - \sec^{4} (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.43.42.1.1.1.3

Kombiniere $2$ und $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ .

$(1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + \tan^{4} (x) - \sec^{4} (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.43.42.1.1.1.4

Schreibe $\tan (x)$ mithilfe von Sinus und Kosinus um.

$(1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + {(\frac{\sin (x)}{\cos (x)})}^{4} - \sec^{4} (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.43.42.1.1.1.5

Wende die Produktregel auf $\frac{\sin (x)}{\cos (x)}$ an.

$(1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + \frac{\sin^{4} (x)}{\cos^{4} (x)} - \sec^{4} (x)) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.43.42.1.1.1.6

Schreibe $\sec (x)$ mithilfe von Sinus und Kosinus um.

$(1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + \frac{\sin^{4} (x)}{\cos^{4} (x)} - {(\frac{1}{\cos (x)})}^{4}) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.43.42.1.1.1.7

Wende die Produktregel auf $\frac{1}{\cos (x)}$ an.

$(1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + \frac{\sin^{4} (x)}{\cos^{4} (x)} - \frac{1^{4}}{\cos^{4} (x)}) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.43.42.1.1.1.8

Eins zu einer beliebigen Potenz erhoben ergibt eins.

$(1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + \frac{\sin^{4} (x)}{\cos^{4} (x)} - \frac{1}{\cos^{4} (x)}) \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.43.42.1.1.2

Wende das Distributivgesetz an.

$1 \cos (x) + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} \cos (x) + \frac{\sin^{4} (x)}{\cos^{4} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.43.42.1.1.3

Vereinfache.

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Schritt 53.43.43.42.1.1.3.1

Mutltipliziere $\cos (x)$ mit $1$ .

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} \cos (x) + \frac{\sin^{4} (x)}{\cos^{4} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.43.42.1.1.3.2

Kürze den gemeinsamen Faktor von $\cos (x)$ .

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Schritt 53.43.43.42.1.1.3.2.1

Faktorisiere $\cos (x)$ aus $\cos^{2} (x)$ heraus.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x) \cos (x)} \cos (x) + \frac{\sin^{4} (x)}{\cos^{4} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.43.42.1.1.3.2.2

Kürze den gemeinsamen Faktor.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x) \cos (x)} \cos (x) + \frac{\sin^{4} (x)}{\cos^{4} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.43.42.1.1.3.2.3

Forme den Ausdruck um.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{4} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.43.42.1.1.3.3

Kürze den gemeinsamen Faktor von $\cos (x)$ .

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Schritt 53.43.43.42.1.1.3.3.1

Faktorisiere $\cos (x)$ aus $\cos^{4} (x)$ heraus.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos (x) \cos^{3} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.43.42.1.1.3.3.2

Kürze den gemeinsamen Faktor.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos (x) \cos^{3} (x)} \cos (x) - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.43.42.1.1.3.3.3

Forme den Ausdruck um.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.43.42.1.1.3.4

Kürze den gemeinsamen Faktor von $\cos (x)$ .

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Schritt 53.43.43.42.1.1.3.4.1

Bringe das führende Minuszeichen in $- \frac{1}{\cos^{4} (x)}$ in den Zähler.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} + \frac{- 1}{\cos^{4} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.43.42.1.1.3.4.2

Faktorisiere $\cos (x)$ aus $\cos^{4} (x)$ heraus.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} + \frac{- 1}{\cos (x) \cos^{3} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.43.42.1.1.3.4.3

Kürze den gemeinsamen Faktor.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} + \frac{- 1}{\cos (x) \cos^{3} (x)} \cos (x) = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.43.42.1.1.3.4.4

Forme den Ausdruck um.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} + \frac{- 1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.43.42.1.1.4

Ziehe das Minuszeichen vor den Bruch.

$\cos (x) + \frac{2 \sin^{2} (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.43.42.1.1.5

Vereinfache jeden Term.

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Schritt 53.43.43.42.1.1.5.1

Faktorisiere $\sin (x)$ aus $\sin^{2} (x)$ heraus.

$\cos (x) + \frac{2 (\sin (x) \sin (x))}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.43.42.1.1.5.2

Separiere Brüche.

$\cos (x) + \frac{2 (\sin (x))}{1} \cdot \frac{\sin (x)}{\cos (x)} + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.43.42.1.1.5.3

Wandle von $\frac{\sin (x)}{\cos (x)}$ nach $\tan (x)$ um.

$\cos (x) + \frac{2 (\sin (x))}{1} \tan (x) + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.43.42.1.1.5.4

Dividiere $2 (\sin (x))$ durch $1$ .

$\cos (x) + 2 (\sin (x)) \tan (x) + \frac{\sin^{4} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.43.42.1.1.5.5

Faktorisiere $\sin^{3} (x)$ aus $\sin^{4} (x)$ heraus.

$\cos (x) + 2 \sin (x) \tan (x) + \frac{\sin^{3} (x) \sin (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.43.42.1.1.5.6

Multipliziere mit $1$ .

$\cos (x) + 2 \sin (x) \tan (x) + \frac{\sin^{3} (x) \sin (x)}{\cos^{3} (x) \cdot 1} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.43.42.1.1.5.7

Separiere Brüche.

$\cos (x) + 2 \sin (x) \tan (x) + \frac{\sin (x)}{1} \cdot \frac{\sin^{3} (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.43.42.1.1.5.8

Wandle von $\frac{\sin^{3} (x)}{\cos^{3} (x)}$ nach $\tan^{3} (x)$ um.

$\cos (x) + 2 \sin (x) \tan (x) + \frac{\sin (x)}{1} \tan^{3} (x) - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.43.42.1.1.5.9

Dividiere $\sin (x)$ durch $1$ .

$\cos (x) + 2 \sin (x) \tan (x) + \sin (x) \tan^{3} (x) - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.43.42.2

Vereinfache die rechte Seite.

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Schritt 53.43.43.42.2.1

Kürze den gemeinsamen Faktor von $\cos (x)$ .

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Schritt 53.43.43.42.2.1.1

Kürze den gemeinsamen Faktor.

$\cos (x) + 2 \sin (x) \tan (x) + \sin (x) \tan^{3} (x) - \frac{1}{\cos^{3} (x)} = \frac{0}{\cos (x)} \cos (x)$

Schritt 53.43.43.42.2.1.2

Forme den Ausdruck um.

$\cos (x) + 2 \sin (x) \tan (x) + \sin (x) \tan^{3} (x) - \frac{1}{\cos^{3} (x)} = 0$

Schritt 53.43.43.43

Löse nach $x$ auf.

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Schritt 53.43.43.43.1

Teile jeden Term in der Gleichung durch $\cos (x)$ .

$\frac{\cos (x)}{\cos (x)} + \frac{2 \sin (x) \tan (x)}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.2

Kürze den gemeinsamen Faktor von $\cos (x)$ .

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Schritt 53.43.43.43.2.1

Kürze den gemeinsamen Faktor.

$\frac{\cos (x)}{\cos (x)} + \frac{2 \sin (x) \tan (x)}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.2.2

Forme den Ausdruck um.

$1 + \frac{2 \sin (x) \tan (x)}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.3

Vereinfache den Zähler.

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Schritt 53.43.43.43.3.1

Schreibe $\tan (x)$ mithilfe von Sinus und Kosinus um.

$1 + \frac{2 \sin (x) (\frac{\sin (x)}{\cos (x)})}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.3.2

Kombiniere Exponenten.

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Schritt 53.43.43.43.3.2.1

Kombiniere $\frac{\sin (x)}{\cos (x)}$ und $2$ .

$1 + \frac{\frac{\sin (x) \cdot 2}{\cos (x)} \cdot \sin (x)}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.3.2.2

Kombiniere $\frac{\sin (x) \cdot 2}{\cos (x)}$ und $\sin (x)$ .

$1 + \frac{\frac{\sin (x) \cdot (2 \sin (x))}{\cos (x)}}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.3.2.3

Potenziere $\sin (x)$ mit $1$ .

$1 + \frac{\frac{2 (\sin (x) \sin (x))}{\cos (x)}}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.3.2.4

Potenziere $\sin (x)$ mit $1$ .

$1 + \frac{\frac{2 (\sin (x) \sin (x))}{\cos (x)}}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.3.2.5

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$1 + \frac{\frac{2 {\sin (x)}^{1 + 1}}{\cos (x)}}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.3.2.6

Addiere $1$ und $1$ .

$1 + \frac{\frac{2 \sin^{2} (x)}{\cos (x)}}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.4

Multipliziere den Zähler mit dem Kehrwert des Nenners.

$1 + \frac{2 \sin^{2} (x)}{\cos (x)} \cdot \frac{1}{\cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.5

Kombiniere Brüche.

Tippen, um mehr Schritte zu sehen ...

Schritt 53.43.43.43.5.1

Kombinieren.

$1 + \frac{2 \sin^{2} (x) \cdot 1}{\cos (x) \cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.5.2

Mutltipliziere $2$ mit $1$ .

$1 + \frac{2 \sin^{2} (x)}{\cos (x) \cos (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.6

Vereinfache den Nenner.

Tippen, um mehr Schritte zu sehen ...

Schritt 53.43.43.43.6.1

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$1 + \frac{2 \sin^{2} (x)}{{\cos (x)}^{1 + 1}} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.6.2

Addiere $1$ und $1$ .

$1 + \frac{2 \sin^{2} (x)}{\cos^{2} (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.7

Multipliziere mit $1$ .

$1 + \frac{2 (\sin^{2} (x) \cdot 1)}{\cos^{2} (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.8

Multipliziere mit $1$ .

$1 + \frac{2 (\sin^{2} (x) \cdot 1)}{\cos^{2} (x) \cdot 1} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.9

Separiere Brüche.

$1 + \frac{2 \cdot (1)}{1} \cdot \frac{\sin^{2} (x)}{\cos^{2} (x)} + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.10

Wandle von $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ nach $\tan^{2} (x)$ um.

$1 + \frac{2 \cdot (1)}{1} \cdot \tan^{2} (x) + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.11

Mutltipliziere $2$ mit $1$ .

$1 + \frac{2}{1} \cdot \tan^{2} (x) + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.12

Dividiere $2$ durch $1$ .

$1 + 2 \tan^{2} (x) + \frac{\sin (x) \tan^{3} (x)}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.13

Vereinfache den Zähler.

Tippen, um mehr Schritte zu sehen ...

Schritt 53.43.43.43.13.1

Schreibe $\tan (x)$ mithilfe von Sinus und Kosinus um.

$1 + 2 \tan^{2} (x) + \frac{\sin (x) {(\frac{\sin (x)}{\cos (x)})}^{3}}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.13.2

Wende die Produktregel auf $\frac{\sin (x)}{\cos (x)}$ an.

$1 + 2 \tan^{2} (x) + \frac{\sin (x) (\frac{\sin^{3} (x)}{\cos^{3} (x)})}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.14

Kombiniere $\sin (x)$ und $\frac{\sin^{3} (x)}{\cos^{3} (x)}$ .

$1 + 2 \tan^{2} (x) + \frac{\frac{\sin (x) \sin^{3} (x)}{\cos^{3} (x)}}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.15

Multipliziere $\sin (x)$ mit $\sin^{3} (x)$ durch Addieren der Exponenten.

Tippen, um mehr Schritte zu sehen ...

Schritt 53.43.43.43.15.1

Mutltipliziere $\sin (x)$ mit $\sin^{3} (x)$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 53.43.43.43.15.1.1

Potenziere $\sin (x)$ mit $1$ .

$1 + 2 \tan^{2} (x) + \frac{\frac{\sin (x) \sin^{3} (x)}{\cos^{3} (x)}}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.15.1.2

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$1 + 2 \tan^{2} (x) + \frac{\frac{{\sin (x)}^{1 + 3}}{\cos^{3} (x)}}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.15.2

Addiere $1$ und $3$ .

$1 + 2 \tan^{2} (x) + \frac{\frac{\sin^{4} (x)}{\cos^{3} (x)}}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.16

Multipliziere den Zähler mit dem Kehrwert des Nenners.

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x)}{\cos^{3} (x)} \cdot \frac{1}{\cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.17

Kombinieren.

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x) \cdot 1}{\cos^{3} (x) \cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.18

Multipliziere $\cos^{3} (x)$ mit $\cos (x)$ durch Addieren der Exponenten.

Tippen, um mehr Schritte zu sehen ...

Schritt 53.43.43.43.18.1

Mutltipliziere $\cos^{3} (x)$ mit $\cos (x)$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 53.43.43.43.18.1.1

Potenziere $\cos (x)$ mit $1$ .

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x) \cdot 1}{\cos^{3} (x) \cos (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.18.1.2

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x) \cdot 1}{{\cos (x)}^{3 + 1}} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.18.2

Addiere $3$ und $1$ .

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x) \cdot 1}{\cos^{4} (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.19

Mutltipliziere $\sin^{4} (x)$ mit $1$ .

$1 + 2 \tan^{2} (x) + \frac{\sin^{4} (x)}{\cos^{4} (x)} + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.20

Wandle von $\frac{\sin^{4} (x)}{\cos^{4} (x)}$ nach $\tan^{4} (x)$ um.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) + \frac{- \frac{1}{\cos^{3} (x)}}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.21

Multipliziere den Zähler mit dem Kehrwert des Nenners.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \frac{1}{\cos^{3} (x)} \cdot \frac{1}{\cos (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.22

Wandle von $\frac{1}{\cos (x)}$ nach $\sec (x)$ um.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \frac{1}{\cos^{3} (x)} \cdot \sec (x) = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.23

Kombiniere $\sec (x)$ und $\frac{1}{\cos^{3} (x)}$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \frac{\sec (x)}{\cos^{3} (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.24

Faktorisiere $\cos (x)$ aus $\cos^{3} (x)$ heraus.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \frac{\sec (x)}{\cos (x) \cos^{2} (x)} = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.25

Separiere Brüche.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot \frac{\sec (x)}{\cos (x)}) = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.26

Schreibe $\sec (x)$ mithilfe von Sinus und Kosinus um.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot \frac{\frac{1}{\cos (x)}}{\cos (x)}) = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.27

Schreibe $\frac{\frac{1}{\cos (x)}}{\cos (x)}$ als ein Produkt um.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot (\frac{1}{\cos (x)} \cdot \frac{1}{\cos (x)})) = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.28

Vereinfache.

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Schritt 53.43.43.43.28.1

Wandle von $\frac{1}{\cos (x)}$ nach $\sec (x)$ um.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot (\sec (x) (\frac{1}{\cos (x)}))) = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.28.2

Wandle von $\frac{1}{\cos (x)}$ nach $\sec (x)$ um.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot (\sec (x) \sec (x))) = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.28.3

Potenziere $\sec (x)$ mit $1$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot (\sec (x) \sec (x))) = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.28.4

Potenziere $\sec (x)$ mit $1$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot (\sec (x) \sec (x))) = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.28.5

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot {\sec (x)}^{1 + 1}) = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.28.6

Addiere $1$ und $1$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1}{\cos^{2} (x)} \cdot \sec^{2} (x)) = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.29

Schreibe $1$ als $1^{2}$ um.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\frac{1^{2}}{\cos^{2} (x)} \cdot \sec^{2} (x)) = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.30

Schreibe $\frac{1^{2}}{\cos^{2} (x)}$ als ${(\frac{1}{\cos (x)})}^{2}$ um.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - ({(\frac{1}{\cos (x)})}^{2} \sec^{2} (x)) = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.31

Wandle von $\frac{1}{\cos (x)}$ nach $\sec (x)$ um.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - (\sec^{2} (x) \sec^{2} (x)) = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.32

Multipliziere $\sec^{2} (x)$ mit $\sec^{2} (x)$ durch Addieren der Exponenten.

Tippen, um mehr Schritte zu sehen ...

Schritt 53.43.43.43.32.1

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - {\sec (x)}^{2 + 2} = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.32.2

Addiere $2$ und $2$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = \frac{0}{\cos (x)}$

Schritt 53.43.43.43.33

Separiere Brüche.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = \frac{0}{1} \cdot \frac{1}{\cos (x)}$

Schritt 53.43.43.43.34

Wandle von $\frac{1}{\cos (x)}$ nach $\sec (x)$ um.

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = \frac{0}{1} \cdot \sec (x)$

Schritt 53.43.43.43.35

Dividiere $0$ durch $1$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = 0 \sec (x)$

Schritt 53.43.43.43.36

Mutltipliziere $0$ mit $\sec (x)$ .

$1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = 0$

Schritt 53.43.43.43.37

Vereinfache die linke Seite.

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Schritt 53.43.43.43.37.1

Vereinfache $1 + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x)$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 53.43.43.43.37.1.1

Faktorisiere unter Verwendung der binomischen Formeln.

Tippen, um mehr Schritte zu sehen ...

Schritt 53.43.43.43.37.1.1.1

Schreibe $1$ als $1^{2}$ um.

$1^{2} + 2 \tan^{2} (x) + \tan^{4} (x) - \sec^{4} (x) = 0$

Schritt 53.43.43.43.37.1.1.2

Schreibe $\tan^{4} (x)$ als ${(\tan^{2} (x))}^{2}$ um.

$1^{2} + 2 \tan^{2} (x) + {(\tan^{2} (x))}^{2} - \sec^{4} (x) = 0$

Schritt 53.43.43.43.37.1.1.3

Überprüfe, ob der mittlere Term das Zweifache des Produkts der Zahlen ist, die im ersten Term und im dritten Term quadriert werden.

$2 \tan^{2} (x) = 2 \cdot 1 \cdot \tan^{2} (x)$

Schritt 53.43.43.43.37.1.1.4

Schreibe das Polynom neu.

$1^{2} + 2 \cdot 1 \cdot \tan^{2} (x) + {(\tan^{2} (x))}^{2} - \sec^{4} (x) = 0$

Schritt 53.43.43.43.37.1.1.5

Faktorisiere mithilfe der trinomischen Formel für das perfekte Quadrat $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , wobei $a = 1$ und $b = \tan^{2} (x)$ .

${(1 + \tan^{2} (x))}^{2} - \sec^{4} (x) = 0$

Schritt 53.43.43.43.37.1.2

Schreibe $\sec^{4} (x)$ als ${(\sec^{2} (x))}^{2}$ um.

${(1 + \tan^{2} (x))}^{2} - {(\sec^{2} (x))}^{2} = 0$

Schritt 53.43.43.43.37.1.3

Da beide Terme perfekte Quadrate sind, faktorisiere durch Anwendung der dritten binomischen Formel, $a^{2} - b^{2} = (a + b) (a - b)$ , mit $a = 1 + \tan^{2} (x)$ und $b = \sec^{2} (x)$ .

$(1 + \tan^{2} (x) + \sec^{2} (x)) (1 + \tan^{2} (x) - \sec^{2} (x)) = 0$

Schritt 53.43.43.43.37.1.4

Ordne Terme um.

$(\tan^{2} (x) + 1 + \sec^{2} (x)) (1 + \tan^{2} (x) - \sec^{2} (x)) = 0$

Schritt 53.43.43.43.37.1.5

Wende den trigonometrischen Pythagoras an.

$(\sec^{2} (x) + \sec^{2} (x)) (1 + \tan^{2} (x) - \sec^{2} (x)) = 0$

Schritt 53.43.43.43.37.1.6

Addiere $\sec^{2} (x)$ und $\sec^{2} (x)$ .

$2 \sec^{2} (x) (1 + \tan^{2} (x) - \sec^{2} (x)) = 0$

Schritt 53.43.43.43.37.1.7

Ordne Terme um.

$2 \sec^{2} (x) (\tan^{2} (x) + 1 - \sec^{2} (x)) = 0$

Schritt 53.43.43.43.37.1.8

Wende den trigonometrischen Pythagoras an.

$2 \sec^{2} (x) (\sec^{2} (x) - \sec^{2} (x)) = 0$

Schritt 53.43.43.43.37.1.9

Subtrahiere $\sec^{2} (x)$ von $\sec^{2} (x)$ .

$2 \sec^{2} (x) \cdot 0 = 0$

Schritt 53.43.43.43.37.1.10

Multipliziere $2 \sec^{2} (x) \cdot 0$ .

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Schritt 53.43.43.43.37.1.10.1

Mutltipliziere $0$ mit $2$ .

$0 \sec^{2} (x) = 0$

Schritt 53.43.43.43.37.1.10.2

Mutltipliziere $0$ mit $\sec^{2} (x)$ .

$0 = 0$

Schritt 53.43.43.43.38

Da $0 = 0$ , wird die Gleichung immer erfüllt sein für jeden Wert von $x$ .

Alle reellen Zahlen

Schritt 54

Das Ergebnis kann in mehreren Formen wiedergegeben werden.

Alle reellen Zahlen

Intervallschreibweise:

$(- \infty, \infty)$