Elementarmathematik Beispiele

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

Schritt 1

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

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

Schritt 2

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

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

Schritt 3

Multipliziere $(\frac{\sin (x)}{\cos (x)} + 1) (\frac{\sin (x)}{\cos (x)} + 1)$ aus unter Verwendung der FOIL-Methode.

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Schritt 3.1

Wende das Distributivgesetz an.

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

Schritt 3.2

Wende das Distributivgesetz an.

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

Schritt 3.3

Wende das Distributivgesetz an.

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

Schritt 4

Vereinfache und fasse gleichartige Terme zusammen.

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Schritt 4.1

Vereinfache jeden Term.

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Schritt 4.1.1

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

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Schritt 4.1.1.1

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

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

Schritt 4.1.1.2

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

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

Schritt 4.1.1.3

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

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

Schritt 4.1.1.4

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

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

Schritt 4.1.1.5

Addiere $1$ und $1$ .

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

Schritt 4.1.1.6

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

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

Schritt 4.1.1.7

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

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

Schritt 4.1.1.8

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

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

Schritt 4.1.1.9

Addiere $1$ und $1$ .

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

Schritt 4.1.2

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

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

Schritt 4.1.3

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

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

Schritt 4.1.4

Mutltipliziere $1$ mit $1$ .

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

Schritt 4.2

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

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

Schritt 5

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

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

Schritt 6

Wende das Distributivgesetz an.

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

Schritt 7

Vereinfache.

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Schritt 7.1

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

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Schritt 7.1.1

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

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

Schritt 7.1.2

Kürze den gemeinsamen Faktor.

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

Schritt 7.1.3

Forme den Ausdruck um.

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

Schritt 7.2

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

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Schritt 7.2.1

Kürze den gemeinsamen Faktor.

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

Schritt 7.2.2

Forme den Ausdruck um.

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

Schritt 7.3

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

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

Schritt 8

Vereinfache jeden Term.

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Schritt 8.1

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

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

Schritt 8.2

Separiere Brüche.

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

Schritt 8.3

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

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

Schritt 8.4

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

$\sin (x) \tan (x) + 2 \sin (x) + \cos (x)$