Trigonometrie Beispiele

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

Schritt 1

Subtrahiere ${(\sec (x) - \tan (x))}^{2}$ von beiden Seiten der Gleichung.

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

Schritt 2

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

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Schritt 2.1

Vereinfache jeden Term.

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Schritt 2.1.1

Vereinfache jeden Term.

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Schritt 2.1.1.1

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

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

Schritt 2.1.1.2

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

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

Schritt 2.1.2

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

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

Schritt 2.1.3

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

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Schritt 2.1.3.1

Wende das Distributivgesetz an.

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

Schritt 2.1.3.2

Wende das Distributivgesetz an.

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

Schritt 2.1.3.3

Wende das Distributivgesetz an.

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

Schritt 2.1.4

Vereinfache und fasse gleichartige Terme zusammen.

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Schritt 2.1.4.1

Vereinfache jeden Term.

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Schritt 2.1.4.1.1

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

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Schritt 2.1.4.1.1.1

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

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

Schritt 2.1.4.1.1.2

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

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

Schritt 2.1.4.1.1.3

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

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

Schritt 2.1.4.1.1.4

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

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

Schritt 2.1.4.1.1.5

Addiere $1$ und $1$ .

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

Schritt 2.1.4.1.2

Schreibe neu unter Anwendung des Kommutativgesetzes der Multiplikation.

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

Schritt 2.1.4.1.3

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

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Schritt 2.1.4.1.3.1

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

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

Schritt 2.1.4.1.3.2

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

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

Schritt 2.1.4.1.3.3

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

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

Schritt 2.1.4.1.3.4

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

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

Schritt 2.1.4.1.3.5

Addiere $1$ und $1$ .

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

Schritt 2.1.4.1.4

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

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Schritt 2.1.4.1.4.1

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

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

Schritt 2.1.4.1.4.2

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

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

Schritt 2.1.4.1.4.3

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

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

Schritt 2.1.4.1.4.4

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

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

Schritt 2.1.4.1.4.5

Addiere $1$ und $1$ .

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

Schritt 2.1.4.1.5

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

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Schritt 2.1.4.1.5.1

Mutltipliziere $- 1$ mit $- 1$ .

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

Schritt 2.1.4.1.5.2

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

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

Schritt 2.1.4.1.5.3

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

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

Schritt 2.1.4.1.5.4

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

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

Schritt 2.1.4.1.5.5

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

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

Schritt 2.1.4.1.5.6

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

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

Schritt 2.1.4.1.5.7

Addiere $1$ und $1$ .

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

Schritt 2.1.4.1.5.8

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

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

Schritt 2.1.4.1.5.9

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

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

Schritt 2.1.4.1.5.10

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

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

Schritt 2.1.4.1.5.11

Addiere $1$ und $1$ .

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

Schritt 2.1.4.2

Subtrahiere $\frac{\sin (x)}{\cos^{2} (x)}$ von $- \frac{\sin (x)}{\cos^{2} (x)}$ .

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

Schritt 2.1.5

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Schritt 2.1.5.1

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

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

Schritt 2.1.5.2

Ziehe das Minuszeichen vor den Bruch.

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

Schritt 2.1.6

Wende das Distributivgesetz an.

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

Schritt 2.1.7

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

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Schritt 2.1.7.1

Mutltipliziere $- 1$ mit $- 1$ .

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

Schritt 2.1.7.2

Mutltipliziere $\frac{2 \sin (x)}{\cos^{2} (x)}$ mit $1$ .

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

Schritt 2.2

Um $\frac{1 - \sin (x)}{1 + \sin (x)}$ als Bruch mit einem gemeinsamen Nenner zu schreiben, multipliziere mit $\frac{\cos^{2} (x)}{\cos^{2} (x)}$ .

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

Schritt 2.3

Um $- \frac{1}{\cos^{2} (x)}$ als Bruch mit einem gemeinsamen Nenner zu schreiben, multipliziere mit $\frac{1 + \sin (x)}{1 + \sin (x)}$ .

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

Schritt 2.4

Schreibe jeden Ausdruck mit einem gemeinsamen Nenner von $(1 + \sin (x)) \cos^{2} (x)$ , indem du jeden mit einem entsprechenden Faktor von $1$ multiplizierst.

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Schritt 2.4.1

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

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

Schritt 2.4.2

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

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

Schritt 2.4.3

Stelle die Faktoren von $(1 + \sin (x)) \cos^{2} (x)$ um.

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

Schritt 2.5

Vereinige die Zähler über dem gemeinsamen Nenner.

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

Schritt 2.6

Vereinfache jeden Term.

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Schritt 2.6.1

Vereinfache den Zähler.

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Schritt 2.6.1.1

Wende das Distributivgesetz an.

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

Schritt 2.6.1.2

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

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

Schritt 2.6.1.3

Wende das Distributivgesetz an.

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

Schritt 2.6.1.4

Mutltipliziere $- 1$ mit $1$ .

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

Schritt 2.6.1.5

Schreibe $\cos^{2} (x) - \sin (x) \cos^{2} (x) - 1 - \sin (x)$ in eine faktorisierte Form um.

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Schritt 2.6.1.5.1

Gruppiere die Terme um.

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

Schritt 2.6.1.5.2

Füge Klammern hinzu.

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

Schritt 2.6.1.5.3

Es sei $u = \sin (x) \cos^{2} (x)$ . Ersetze $u$ für alle $\sin (x) \cos^{2} (x)$ .

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

Schritt 2.6.1.5.4

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

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Schritt 2.6.1.5.4.1

Stelle $- \sin^{2} (x)$ und $- u$ um.

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

Schritt 2.6.1.5.4.2

Faktorisiere $- 1$ aus $- u$ heraus.

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

Schritt 2.6.1.5.4.3

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

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

Schritt 2.6.1.5.4.4

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

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

Schritt 2.6.1.5.5

Ersetze alle $u$ durch $\sin (x) \cos^{2} (x)$ .

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

Schritt 2.6.1.5.6

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

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Schritt 2.6.1.5.6.1

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

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

Schritt 2.6.1.5.6.2

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

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

Schritt 2.6.1.5.6.3

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

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

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

Schritt 2.6.1.5.7

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

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Schritt 2.6.1.5.7.1

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

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

Schritt 2.6.1.5.7.2

Faktorisiere $- \sin (x)$ aus $- \sin (x) (1) - \sin (x) (\cos^{2} (x) + \sin (x))$ heraus.

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

Schritt 2.6.2

Ziehe das Minuszeichen vor den Bruch.

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

Schritt 2.7

Um $\frac{2 \sin (x)}{\cos^{2} (x)}$ als Bruch mit einem gemeinsamen Nenner zu schreiben, multipliziere mit $\frac{1 + \sin (x)}{1 + \sin (x)}$ .

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

Schritt 2.8

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

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

Schritt 2.9

Vereinige die Zähler über dem gemeinsamen Nenner.

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

Schritt 2.10

Vereinfache jeden Term.

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Schritt 2.10.1

Vereinfache den Zähler.

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Schritt 2.10.1.1

Faktorisiere $\sin (x)$ aus $- \sin (x) (1 + \cos^{2} (x) + \sin (x)) + 2 \sin (x) (1 + \sin (x))$ heraus.

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Schritt 2.10.1.1.1

Faktorisiere $\sin (x)$ aus $- \sin (x) (1 + \cos^{2} (x) + \sin (x))$ heraus.

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

Schritt 2.10.1.1.2

Faktorisiere $\sin (x)$ aus $2 \sin (x) (1 + \sin (x))$ heraus.

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

Schritt 2.10.1.1.3

Faktorisiere $\sin (x)$ aus $\sin (x) (- 1 (1 + \cos^{2} (x) + \sin (x))) + \sin (x) (2 (1 + \sin (x)))$ heraus.

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

Schritt 2.10.1.2

Wende das Distributivgesetz an.

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

Schritt 2.10.1.3

Vereinfache.

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Schritt 2.10.1.3.1

Mutltipliziere $- 1$ mit $1$ .

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

Schritt 2.10.1.3.2

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

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

Schritt 2.10.1.3.3

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

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

Schritt 2.10.1.4

Wende das Distributivgesetz an.

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

Schritt 2.10.1.5

Mutltipliziere $2$ mit $1$ .

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

Schritt 2.10.1.6

Addiere $- 1$ und $2$ .

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

Schritt 2.10.1.7

Addiere $- \sin (x)$ und $2 \sin (x)$ .

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

Schritt 2.10.1.8

Wende den trigonometrischen Pythagoras an.

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

Schritt 2.10.1.9

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

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Schritt 2.10.1.9.1

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

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

Schritt 2.10.1.9.2

Multipliziere mit $1$ .

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

Schritt 2.10.1.9.3

Faktorisiere $\sin (x)$ aus $\sin (x) \sin (x) + \sin (x) \cdot 1$ heraus.

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

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

Schritt 2.10.1.10

Kombiniere Exponenten.

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Schritt 2.10.1.10.1

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

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

Schritt 2.10.1.10.2

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

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

Schritt 2.10.1.10.3

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

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

Schritt 2.10.1.10.4

Addiere $1$ und $1$ .

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

Schritt 2.10.2

Kürze den gemeinsamen Teiler von $\sin (x) + 1$ und $1 + \sin (x)$ .

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Schritt 2.10.2.1

Stelle die Terme um.

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

Schritt 2.10.2.2

Kürze den gemeinsamen Faktor.

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

Schritt 2.10.2.3

Forme den Ausdruck um.

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

Schritt 2.11

Subtrahiere $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ von $\frac{\sin^{2} (x)}{\cos^{2} (x)}$ .

$0 = 0$

Schritt 3

Der Definitionsbereich umfasst alle reellen Zahlen, ausgenommen jene, für die der Ausdruck nicht definiert ist. In diesem Fall gibt es keine reellen Zahlen, für die der Ausdruck nicht definiert ist.