Elementarmathematik Beispiele

$\frac{\cot (x) - \tan (x)}{\cot (x) + \tan (x)} = \cos (2 x)$

Schritt 1

Beginne auf der linken Seite.

$\frac{\cot (x) - \tan (x)}{\cot (x) + \tan (x)}$

Schritt 2

Vereinfache den Ausdruck.

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Schritt 2.1

Vereinfache den Zähler.

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Schritt 2.1.1

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

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

Schritt 2.1.2

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

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

Schritt 2.2

Vereinfache den Nenner.

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Schritt 2.2.1

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

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

Schritt 2.2.2

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

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

Schritt 2.3

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

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Schritt 2.3.1

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

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

Schritt 2.3.2

Kombinieren.

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

Schritt 2.4

Wende das Distributivgesetz an.

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

Schritt 2.5

Vereinfache durch Kürzen.

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Schritt 2.5.1

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

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Schritt 2.5.1.1

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

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

Schritt 2.5.1.2

Kürze den gemeinsamen Faktor.

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

Schritt 2.5.1.3

Forme den Ausdruck um.

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

Schritt 2.5.2

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

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

Schritt 2.5.3

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

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

Schritt 2.5.4

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

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

Schritt 2.5.5

Addiere $1$ und $1$ .

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

Schritt 2.5.6

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

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Schritt 2.5.6.1

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

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

Schritt 2.5.6.2

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

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

Schritt 2.5.6.3

Kürze den gemeinsamen Faktor.

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

Schritt 2.5.6.4

Forme den Ausdruck um.

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

Schritt 2.5.7

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

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

Schritt 2.5.8

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

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

Schritt 2.5.9

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

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

Schritt 2.5.10

Addiere $1$ und $1$ .

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

Schritt 2.5.11

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

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Schritt 2.5.11.1

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

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

Schritt 2.5.11.2

Kürze den gemeinsamen Faktor.

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

Schritt 2.5.11.3

Forme den Ausdruck um.

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

Schritt 2.5.12

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

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

Schritt 2.5.13

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

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

Schritt 2.5.14

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

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

Schritt 2.5.15

Addiere $1$ und $1$ .

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

Schritt 2.5.16

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

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Schritt 2.5.16.1

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

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

Schritt 2.5.16.2

Kürze den gemeinsamen Faktor.

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

Schritt 2.5.16.3

Forme den Ausdruck um.

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

Schritt 2.5.17

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

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

Schritt 2.5.18

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

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

Schritt 2.5.19

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

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

Schritt 2.5.20

Addiere $1$ und $1$ .

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

Schritt 2.6

Ordne Terme um.

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

Schritt 2.7

Wende den trigonometrischen Pythagoras an.

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

Schritt 2.8

Dividiere $\cos^{2} (x) - \sin^{2} (x)$ durch $1$ .

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

Schritt 2.9

Wende die Doppelwinkelfunktion für den Kosinus an.

$\cos (2 x)$

Schritt 3

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

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