Trigonometrie Beispiele

$\frac{\tan (a) - \tan (b)}{\cot (b) - \cot (a)} = \frac{\tan (b)}{\cot (a)}$

Schritt 1

Beginne auf der linken Seite.

$\frac{\tan (a) - \tan (b)}{\cot (b) - \cot (a)}$

Schritt 2

Wandle in Sinus und Kosinus um.

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Schritt 2.1

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

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

Schritt 2.2

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

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

Schritt 2.3

Schreibe $\cot (b)$ mit Sinus und Kosinus mithilfe der Quotienten-Identitätsgleichung.

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

Schritt 2.4

Schreibe $\cot (a)$ mit Sinus und Kosinus mithilfe der Quotienten-Identitätsgleichung.

$\frac{\frac{\sin (a)}{\cos (a)} - \frac{\sin (b)}{\cos (b)}}{\frac{\cos (b)}{\sin (b)} - \frac{\cos (a)}{\sin (a)}}$

Schritt 3

Vereinfache.

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Schritt 3.1

Multiply the numerator and denominator of the fraction by $\cos (a) \cos (b) \sin (b) \sin (a)$ .

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Schritt 3.1.1

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

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

Schritt 3.1.2

Kombinieren.

$\frac{\cos (a) \cos (b) \sin (b) \sin (a) (\frac{\sin (a)}{\cos (a)} - \frac{\sin (b)}{\cos (b)})}{\cos (a) \cos (b) \sin (b) \sin (a) (\frac{\cos (b)}{\sin (b)} - \frac{\cos (a)}{\sin (a)})}$

Schritt 3.2

Wende das Distributivgesetz an.

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

Schritt 3.3

Vereinfache durch Kürzen.

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Schritt 3.3.1

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

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Schritt 3.3.1.1

Faktorisiere $\cos (a)$ aus $\cos (a) \cos (b) \sin (b) \sin (a)$ heraus.

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

Schritt 3.3.1.2

Kürze den gemeinsamen Faktor.

Schritt 3.3.1.3

Forme den Ausdruck um.

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

Schritt 3.3.2

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

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

Schritt 3.3.3

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

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

Schritt 3.3.4

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

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

Schritt 3.3.5

Addiere $1$ und $1$ .

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

Schritt 3.3.6

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

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Schritt 3.3.6.1

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

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

Schritt 3.3.6.2

Faktorisiere $\cos (b)$ aus $\cos (a) \cos (b) \sin (b) \sin (a)$ heraus.

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

Schritt 3.3.6.3

Kürze den gemeinsamen Faktor.

Schritt 3.3.6.4

Forme den Ausdruck um.

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

Schritt 3.3.7

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

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

Schritt 3.3.8

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

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

Schritt 3.3.9

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

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

Schritt 3.3.10

Addiere $1$ und $1$ .

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

Schritt 3.3.11

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

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Schritt 3.3.11.1

Faktorisiere $\sin (b)$ aus $\cos (a) \cos (b) \sin (b) \sin (a)$ heraus.

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

Schritt 3.3.11.2

Kürze den gemeinsamen Faktor.

Schritt 3.3.11.3

Forme den Ausdruck um.

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

Schritt 3.3.12

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

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

Schritt 3.3.13

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

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

Schritt 3.3.14

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

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

Schritt 3.3.15

Addiere $1$ und $1$ .

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

Schritt 3.3.16

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

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Schritt 3.3.16.1

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

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

Schritt 3.3.16.2

Faktorisiere $\sin (a)$ aus $\cos (a) \cos (b) \sin (b) \sin (a)$ heraus.

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

Schritt 3.3.16.3

Kürze den gemeinsamen Faktor.

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

Schritt 3.3.16.4

Forme den Ausdruck um.

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

Schritt 3.3.17

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

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

Schritt 3.3.18

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

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

Schritt 3.3.19

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

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

Schritt 3.3.20

Addiere $1$ und $1$ .

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

Schritt 3.4

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

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Schritt 3.4.1

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

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

Schritt 3.4.2

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

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

Schritt 3.4.3

Faktorisiere $\sin (b) \sin (a)$ aus $\sin (b) \sin (a) (\cos (b) \sin (a)) + \sin (b) \sin (a) (\cos (a) (- \sin (b)))$ heraus.

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

Schritt 3.5

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

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Schritt 3.5.1

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

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

Schritt 3.5.2

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

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

Schritt 3.5.3

Faktorisiere $\cos (a) \cos (b)$ aus $\cos (a) \cos (b) (\cos (b) \sin (a)) + \cos (a) \cos (b) (\sin (b) (- \cos (a)))$ heraus.

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

Schritt 3.6

Stelle die Terme um.

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

Schritt 3.7

Kürze den gemeinsamen Teiler von $\cos (b) \sin (a) - 1 \cdot \cos (a) \sin (b)$ und $\cos (b) \sin (a) - 1 \cdot \sin (b) \cos (a)$ .

$\frac{\sin (b) \sin (a)}{\cos (a) \cos (b)}$

Schritt 4

Stelle die Terme um.

$\frac{\sin (a) \sin (b)}{\cos (a) \cos (b)}$

Schritt 5

Schreibe $\frac{\sin (a) \sin (b)}{\cos (a) \cos (b)}$ als $\frac{\tan (b)}{\cot (a)}$ um.

$\frac{\tan (b)}{\cot (a)}$

Schritt 6

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

$\frac{\tan (a) - \tan (b)}{\cot (b) - \cot (a)} = \frac{\tan (b)}{\cot (a)}$ ist eine Identitätsgleichung