Elementarmathematik Beispiele

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

Schritt 1

Beginne auf der linken Seite.

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

Schritt 2

Vereinfache den Ausdruck.

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Schritt 2.1

Vereinfache den Zähler.

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Schritt 2.1.1

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

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

Schritt 2.1.2

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

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

Schritt 2.2

Vereinfache den Nenner.

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Schritt 2.2.1

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

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

Schritt 2.2.2

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

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

Schritt 2.3

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

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Schritt 2.3.1

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

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

Schritt 2.3.2

Kombinieren.

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

Schritt 2.4

Wende das Distributivgesetz an.

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

Schritt 2.5

Vereinfache durch Kürzen.

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Schritt 2.5.1

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

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Schritt 2.5.1.1

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

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

Schritt 2.5.1.2

Kürze den gemeinsamen Faktor.

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

Schritt 2.5.1.3

Forme den Ausdruck um.

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

Schritt 2.5.2

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

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

Schritt 2.5.3

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

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

Schritt 2.5.4

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

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

Schritt 2.5.5

Addiere $1$ und $1$ .

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

Schritt 2.5.6

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

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Schritt 2.5.6.1

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

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

Schritt 2.5.6.2

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

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

Schritt 2.5.6.3

Kürze den gemeinsamen Faktor.

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

Schritt 2.5.6.4

Forme den Ausdruck um.

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

Schritt 2.5.7

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

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

Schritt 2.5.8

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

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

Schritt 2.5.9

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

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

Schritt 2.5.10

Addiere $1$ und $1$ .

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

Schritt 2.5.11

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

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Schritt 2.5.11.1

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

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

Schritt 2.5.11.2

Kürze den gemeinsamen Faktor.

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

Schritt 2.5.11.3

Forme den Ausdruck um.

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

Schritt 2.5.12

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

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

Schritt 2.5.13

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

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

Schritt 2.5.14

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

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

Schritt 2.5.15

Addiere $1$ und $1$ .

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

Schritt 2.5.16

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

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Schritt 2.5.16.1

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

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

Schritt 2.5.16.2

Kürze den gemeinsamen Faktor.

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

Schritt 2.5.16.3

Forme den Ausdruck um.

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

Schritt 2.5.17

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

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

Schritt 2.5.18

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

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

Schritt 2.5.19

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

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

Schritt 2.5.20

Addiere $1$ und $1$ .

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

Schritt 2.6

Wende den trigonometrischen Pythagoras an.

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

Schritt 2.7

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

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

Schritt 3

Wende den umgekehrten trigonometrischen Pythagoras an.

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

Schritt 4

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

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

Schritt 5

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

$\frac{\tan (x) - \cot (x)}{\tan (x) + \cot (x)} = 1 - 2 \cos^{2} (x)$ ist eine Identitätsgleichung