Trigonometrie Beispiele

$\frac{\cos (a)}{1 - \tan (a)} + \frac{\sin (a)}{1 - \cot (a)}$

Schritt 1

Vereinfache jeden Term.

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Schritt 1.1

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

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

Schritt 1.2

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

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

Schritt 2

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

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

Schritt 3

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

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

Schritt 4

Schreibe jeden Ausdruck mit einem gemeinsamen Nenner von $(1 - \frac{\sin (a)}{\cos (a)}) (1 - \frac{\cos (a)}{\sin (a)})$ , indem du jeden mit einem entsprechenden Faktor von $1$ multiplizierst.

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Schritt 4.1

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

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

Schritt 4.2

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

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

Schritt 4.3

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

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

Schritt 5

Vereinige die Zähler über dem gemeinsamen Nenner.

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

Schritt 6

Vereinfache den Zähler.

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Schritt 6.1

Wende das Distributivgesetz an.

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

Schritt 6.2

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

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

Schritt 6.3

Schreibe neu unter Anwendung des Kommutativgesetzes der Multiplikation.

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

Schritt 6.4

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

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Schritt 6.4.1

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

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

Schritt 6.4.2

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

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

Schritt 6.4.3

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

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

Schritt 6.4.4

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

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

Schritt 6.4.5

Addiere $1$ und $1$ .

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

Schritt 6.5

Wende das Distributivgesetz an.

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

Schritt 6.6

Mutltipliziere $\sin (a)$ mit $1$ .

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

Schritt 6.7

Schreibe neu unter Anwendung des Kommutativgesetzes der Multiplikation.

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

Schritt 6.8

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

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Schritt 6.8.1

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

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

Schritt 6.8.2

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

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

Schritt 6.8.3

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

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

Schritt 6.8.4

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

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

Schritt 6.8.5

Addiere $1$ und $1$ .

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

Schritt 7

Vereinfache den Zähler.

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Schritt 7.1

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

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

Schritt 7.2

Separiere Brüche.

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

Schritt 7.3

Wandle von $\frac{\cos (a)}{\sin (a)}$ nach $\cot (a)$ um.

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

Schritt 7.4

Dividiere $\cos (a)$ durch $1$ .

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

Schritt 7.5

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

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

Schritt 7.6

Separiere Brüche.

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

Schritt 7.7

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

$\frac{\cos (a) - \cos (a) \cot (a) + \sin (a) - (\frac{\sin (a)}{1} \tan (a))}{(1 - \frac{\cos (a)}{\sin (a)}) (1 - \frac{\sin (a)}{\cos (a)})}$

Schritt 7.8

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

$\frac{\cos (a) - \cos (a) \cot (a) + \sin (a) - \sin (a) \tan (a)}{(1 - \frac{\cos (a)}{\sin (a)}) (1 - \frac{\sin (a)}{\cos (a)})}$

Schritt 8

Vereinfache den Nenner.

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Schritt 8.1

Wandle von $\frac{\cos (a)}{\sin (a)}$ nach $\cot (a)$ um.

$\frac{\cos (a) - \cos (a) \cot (a) + \sin (a) - \sin (a) \tan (a)}{(1 - \cot (a)) (1 - \frac{\sin (a)}{\cos (a)})}$

Schritt 8.2

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

$\frac{\cos (a) - \cos (a) \cot (a) + \sin (a) - \sin (a) \tan (a)}{(1 - \cot (a)) (1 - \tan (a))}$