Trigonometrie Beispiele

$\frac{1 - \sec (x)}{\tan (x)} - \frac{\tan (x)}{1 - \sec (x)} = 2 \cot (x)$

Schritt 1

Beginne auf der linken Seite.

$\frac{1 - \sec (x)}{\tan (x)} - \frac{\tan (x)}{1 - \sec (x)}$

Schritt 2

Mutltipliziere $\frac{\tan (x)}{1 - \sec (x)}$ mit $\frac{1 + \sec (x)}{1 + \sec (x)}$ .

$\frac{1 - \sec (x)}{\tan (x)} - (\frac{\tan (x)}{1 - \sec (x)} \cdot \frac{1 + \sec (x)}{1 + \sec (x)})$

Schritt 3

Kombinieren.

$\frac{1 - \sec (x)}{\tan (x)} - \frac{\tan (x) (1 + \sec (x))}{(1 - \sec (x)) (1 + \sec (x))}$

Schritt 4

Vereinfache den Zähler.

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Schritt 4.1

Wende das Distributivgesetz an.

$\frac{1 - \sec (x)}{\tan (x)} - \frac{\tan (x) \cdot 1 + \tan (x) \sec (x)}{(1 - \sec (x)) (1 + \sec (x))}$

Schritt 4.2

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

$\frac{1 - \sec (x)}{\tan (x)} - \frac{\tan (x) + \tan (x) \sec (x)}{(1 - \sec (x)) (1 + \sec (x))}$

Schritt 5

Vereinfache Nenner.

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Schritt 5.1

Multipliziere $(1 - \sec (x)) (1 + \sec (x))$ aus unter Verwendung der FOIL-Methode.

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Schritt 5.1.1

Wende das Distributivgesetz an.

$\frac{1 - \sec (x)}{\tan (x)} - \frac{\tan (x) + \tan (x) \sec (x)}{1 (1 + \sec (x)) - \sec (x) (1 + \sec (x))}$

Schritt 5.1.2

Wende das Distributivgesetz an.

$\frac{1 - \sec (x)}{\tan (x)} - \frac{\tan (x) + \tan (x) \sec (x)}{1 \cdot 1 + 1 \sec (x) - \sec (x) (1 + \sec (x))}$

Schritt 5.1.3

Wende das Distributivgesetz an.

$\frac{1 - \sec (x)}{\tan (x)} - \frac{\tan (x) + \tan (x) \sec (x)}{1 \cdot 1 + 1 \sec (x) - \sec (x) \cdot 1 - \sec (x) \sec (x)}$

Schritt 5.2

Vereinfache und fasse gleichartige Terme zusammen.

$\frac{1 - \sec (x)}{\tan (x)} - \frac{\tan (x) + \tan (x) \sec (x)}{1 - \sec^{2} (x)}$

Schritt 6

Schreibe $- 1$ als einen Bruch mit dem Nenner $1$ .

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

Schritt 7

Kombinieren.

$\frac{1 - \sec (x)}{\tan (x)} + \frac{- (\tan (x) + \tan (x) \sec (x))}{1 (1 - \sec^{2} (x))}$

Schritt 8

Wende das Distributivgesetz an.

$\frac{1 - \sec (x)}{\tan (x)} + \frac{- \tan (x) - \tan (x) \sec (x)}{1 (1 - \sec^{2} (x))}$

Schritt 9

Mutltipliziere $1 - {\sec (x)}^{2}$ mit $1$ .

$\frac{1 - \sec (x)}{\tan (x)} + \frac{- \tan (x) - \tan (x) \sec (x)}{1 - \sec^{2} (x)}$

Schritt 10

Wende den trigonometrischen Pythagoras an.

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Schritt 10.1

Stelle $1$ und $- \sec^{2} (x)$ um.

$\frac{1 - \sec (x)}{\tan (x)} + \frac{- \tan (x) - \tan (x) \sec (x)}{- \sec^{2} (x) + 1}$

Schritt 10.2

Faktorisiere $- 1$ aus $- \sec^{2} (x)$ heraus.

$\frac{1 - \sec (x)}{\tan (x)} + \frac{- \tan (x) - \tan (x) \sec (x)}{- (\sec^{2} (x)) + 1}$

Schritt 10.3

Schreibe $1$ als $- 1 (- 1)$ um.

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

Schritt 10.4

Faktorisiere $- 1$ aus $- \sec^{2} (x) - 1 \cdot - 1$ heraus.

$\frac{1 - \sec (x)}{\tan (x)} + \frac{- \tan (x) - \tan (x) \sec (x)}{- (\sec^{2} (x) - 1)}$

Schritt 10.5

Wende den trigonometrischen Pythagoras an.

$\frac{1 - \sec (x)}{\tan (x)} + \frac{- \tan (x) - \tan (x) \sec (x)}{- \tan^{2} (x)}$

Schritt 11

Wandle in Sinus und Kosinus um.

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Schritt 11.1

Wende die Kehrwertfunktion auf $\sec (x)$ an.

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

Schritt 11.2

Schreibe $\tan (x)$ mit Sinus und Kosinus mithilfe der Quotienten-Identitätsgleichung.

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

Schritt 11.3

Schreibe $\tan (x)$ mit Sinus und Kosinus mithilfe der Quotienten-Identitätsgleichung.

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

Schritt 11.4

Schreibe $\tan (x)$ mit Sinus und Kosinus mithilfe der Quotienten-Identitätsgleichung.

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

Schritt 11.5

Wende die Kehrwertfunktion auf $\sec (x)$ an.

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

Schritt 11.6

Schreibe $\tan (x)$ mit Sinus und Kosinus mithilfe der Quotienten-Identitätsgleichung.

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

Schritt 11.7

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

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

Schritt 12

Vereinfache.

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Schritt 12.1

Stelle die Faktoren von $- \frac{\sin (x)}{\cos (x)} \cdot \frac{1}{\cos (x)}$ um.

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

Schritt 12.2

Um $- \frac{1}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}$ als Bruch mit einem gemeinsamen Nenner zu schreiben, multipliziere mit $\frac{\cos (x)}{\cos (x)}$ .

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

Schritt 12.3

Kombiniere $- \frac{1}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}$ und $\frac{\cos (x)}{\cos (x)}$ .

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

Schritt 12.4

Vereinige die Zähler über dem gemeinsamen Nenner.

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

Schritt 12.5

Vereinfache jeden Term.

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Schritt 12.5.1

Multipliziere den Zähler mit dem Kehrwert des Nenners.

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

Schritt 12.5.2

Wende das Distributivgesetz an.

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

Schritt 12.5.3

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

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

Schritt 12.5.4

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

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Schritt 12.5.4.1

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

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

Schritt 12.5.4.2

Kürze den gemeinsamen Faktor.

Schritt 12.5.4.3

Forme den Ausdruck um.

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

Schritt 12.5.5

Schreibe $- 1 \frac{1}{\sin (x)}$ als $- \frac{1}{\sin (x)}$ um.

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

Schritt 12.5.6

Multipliziere den Zähler mit dem Kehrwert des Nenners.

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

Schritt 12.5.7

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

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Schritt 12.5.7.1

Bringe das führende Minuszeichen in $- \frac{{\cos (x)}^{2}}{{\sin (x)}^{2}}$ in den Zähler.

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

Schritt 12.5.7.2

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

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

Schritt 12.5.7.3

Kürze den gemeinsamen Faktor.

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

Schritt 12.5.7.4

Forme den Ausdruck um.

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

Schritt 12.5.8

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

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

Schritt 12.5.9

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

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

Schritt 12.5.10

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

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

Schritt 12.5.11

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

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

Schritt 12.5.12

Addiere $1$ und $1$ .

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

Schritt 12.5.13

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

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

Schritt 12.5.14

Kürze den gemeinsamen Teiler von $\cos (x)$ und ${\cos (x)}^{2}$ .

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Schritt 12.5.14.1

Faktorisiere $\cos (x)$ aus $\cos (x) \sin (x)$ heraus.

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

Schritt 12.5.14.2

Kürze die gemeinsamen Faktoren.

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Schritt 12.5.14.2.1

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

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

Schritt 12.5.14.2.2

Kürze den gemeinsamen Faktor.

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

Schritt 12.5.14.2.3

Forme den Ausdruck um.

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

Schritt 12.5.15

Ziehe das Minuszeichen vor den Bruch.

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

Schritt 12.5.16

Wende das Distributivgesetz an.

$\frac{\cos (x)}{\sin (x)} - \frac{1}{\sin (x)} - \sin (x) (- \frac{\cos (x)}{{\sin (x)}^{2}}) - \frac{\sin (x)}{\cos (x)} (- \frac{\cos (x)}{{\sin (x)}^{2}})$

Schritt 12.5.17

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

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Schritt 12.5.17.1

Bringe das führende Minuszeichen in $- \frac{\cos (x)}{{\sin (x)}^{2}}$ in den Zähler.

$\frac{\cos (x)}{\sin (x)} - \frac{1}{\sin (x)} - \sin (x) \frac{- \cos (x)}{{\sin (x)}^{2}} - \frac{\sin (x)}{\cos (x)} (- \frac{\cos (x)}{{\sin (x)}^{2}})$

Schritt 12.5.17.2

Faktorisiere $\sin (x)$ aus $- \sin (x)$ heraus.

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

Schritt 12.5.17.3

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

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

Schritt 12.5.17.4

Kürze den gemeinsamen Faktor.

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

Schritt 12.5.17.5

Forme den Ausdruck um.

$\frac{\cos (x)}{\sin (x)} - \frac{1}{\sin (x)} - 1 \frac{- \cos (x)}{\sin (x)} - \frac{\sin (x)}{\cos (x)} (- \frac{\cos (x)}{{\sin (x)}^{2}})$

Schritt 12.5.18

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

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Schritt 12.5.18.1

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

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

Schritt 12.5.18.2

Bringe das führende Minuszeichen in $- \frac{\cos (x)}{{\sin (x)}^{2}}$ in den Zähler.

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

Schritt 12.5.18.3

Faktorisiere $\sin (x)$ aus $- \sin (x)$ heraus.

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

Schritt 12.5.18.4

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

$\frac{\cos (x)}{\sin (x)} - \frac{1}{\sin (x)} - 1 \frac{- \cos (x)}{\sin (x)} + \frac{\sin (x) \cdot - 1}{\cos (x)} \cdot \frac{- \cos (x)}{\sin (x) \sin (x)}$

Schritt 12.5.18.5

Kürze den gemeinsamen Faktor.

$\frac{\cos (x)}{\sin (x)} - \frac{1}{\sin (x)} - 1 \frac{- \cos (x)}{\sin (x)} + \frac{\sin (x) \cdot - 1}{\cos (x)} \cdot \frac{- \cos (x)}{\sin (x) \sin (x)}$

Schritt 12.5.18.6

Forme den Ausdruck um.

$\frac{\cos (x)}{\sin (x)} - \frac{1}{\sin (x)} - 1 \frac{- \cos (x)}{\sin (x)} + \frac{- 1}{\cos (x)} \cdot \frac{- \cos (x)}{\sin (x)}$

Schritt 12.5.19

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

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Schritt 12.5.19.1

Faktorisiere $\cos (x)$ aus $- \cos (x)$ heraus.

$\frac{\cos (x)}{\sin (x)} - \frac{1}{\sin (x)} - 1 \frac{- \cos (x)}{\sin (x)} + \frac{- 1}{\cos (x)} \cdot \frac{\cos (x) \cdot - 1}{\sin (x)}$

Schritt 12.5.19.2

Kürze den gemeinsamen Faktor.

$\frac{\cos (x)}{\sin (x)} - \frac{1}{\sin (x)} - 1 \frac{- \cos (x)}{\sin (x)} + \frac{- 1}{\cos (x)} \cdot \frac{\cos (x) \cdot - 1}{\sin (x)}$

Schritt 12.5.19.3

Forme den Ausdruck um.

$\frac{\cos (x)}{\sin (x)} - \frac{1}{\sin (x)} - 1 \frac{- \cos (x)}{\sin (x)} - 1 \frac{- 1}{\sin (x)}$

Schritt 12.5.20

Vereinfache jeden Term.

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Schritt 12.5.20.1

Ziehe das Minuszeichen vor den Bruch.

$\frac{\cos (x)}{\sin (x)} - \frac{1}{\sin (x)} - 1 (- \frac{\cos (x)}{\sin (x)}) - 1 \frac{- 1}{\sin (x)}$

Schritt 12.5.20.2

Multipliziere $- 1 (- \frac{\cos (x)}{\sin (x)})$ .

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Schritt 12.5.20.2.1

Mutltipliziere $- 1$ mit $- 1$ .

$\frac{\cos (x)}{\sin (x)} - \frac{1}{\sin (x)} + 1 \frac{\cos (x)}{\sin (x)} - 1 \frac{- 1}{\sin (x)}$

Schritt 12.5.20.2.2

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

$\frac{\cos (x)}{\sin (x)} - \frac{1}{\sin (x)} + \frac{\cos (x)}{\sin (x)} - 1 \frac{- 1}{\sin (x)}$

Schritt 12.5.20.3

Ziehe das Minuszeichen vor den Bruch.

$\frac{\cos (x)}{\sin (x)} - \frac{1}{\sin (x)} + \frac{\cos (x)}{\sin (x)} - 1 (- \frac{1}{\sin (x)})$

Schritt 12.5.20.4

Multipliziere $- 1 (- \frac{1}{\sin (x)})$ .

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Schritt 12.5.20.4.1

Mutltipliziere $- 1$ mit $- 1$ .

$\frac{\cos (x)}{\sin (x)} - \frac{1}{\sin (x)} + \frac{\cos (x)}{\sin (x)} + 1 \frac{1}{\sin (x)}$

Schritt 12.5.20.4.2

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

$\frac{\cos (x)}{\sin (x)} - \frac{1}{\sin (x)} + \frac{\cos (x)}{\sin (x)} + \frac{1}{\sin (x)}$

Schritt 12.6

Vereinige die Zähler über dem gemeinsamen Nenner.

$\frac{\cos (x) - 1 + \cos (x) + 1}{\sin (x)}$

Schritt 12.7

Addiere $\cos (x)$ und $\cos (x)$ .

$\frac{2 \cos (x) - 1 + 1}{\sin (x)}$

Schritt 12.8

Addiere $- 1$ und $1$ .

$\frac{2 \cos (x) + 0}{\sin (x)}$

Schritt 12.9

Addiere $2 \cos (x)$ und $0$ .

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

Schritt 13

Schreibe $\frac{2 \cos (x)}{\sin (x)}$ als $2 \cot (x)$ um.

$2 \cot (x)$

Schritt 14

Da gezeigt wurde, dass die beiden Seiten äquivalent sind, ist die Gleichung eine Identitätsgleichung.

$\frac{1 - \sec (x)}{\tan (x)} - \frac{\tan (x)}{1 - \sec (x)} = 2 \cot (x)$ ist eine Identitätsgleichung