Trigonometrie Beispiele

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

Schritt 1

Beginne auf der linken Seite.

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

Schritt 2

Wandle in Sinus und Kosinus um.

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Schritt 2.1

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

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

Schritt 2.2

Wende die Produktregel auf $\frac{\sin (x)}{\cos (x)}$ an.

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

Schritt 3

Vereinfache.

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Schritt 3.1

Multiply the numerator and denominator of the fraction by ${\cos (x)}^{2}$ .

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Schritt 3.1.1

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

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

Schritt 3.1.2

Kombinieren.

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

Schritt 3.2

Wende das Distributivgesetz an.

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

Schritt 3.3

Kürze den gemeinsamen Faktor von ${\cos (x)}^{2}$ .

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Schritt 3.3.1

Bringe das führende Minuszeichen in $- \frac{{\sin (x)}^{2}}{{\cos (x)}^{2}}$ in den Zähler.

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

Schritt 3.3.2

Kürze den gemeinsamen Faktor.

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

Schritt 3.3.3

Forme den Ausdruck um.

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

Schritt 3.4

Vereinfache den Zähler.

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Schritt 3.4.1

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

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

Schritt 3.4.2

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

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

Schritt 3.4.3

Da beide Terme perfekte Quadrate sind, faktorisiere durch Anwendung der dritten binomischen Formel, $a^{2} - b^{2} = (a + b) (a - b)$ , mit $a = \cos (x) \cos (x)$ und $b = \cos (x) \sin (x)$ .

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

Schritt 3.4.4

Vereinfache.

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Schritt 3.4.4.1

Multipliziere $\cos (x) \cos (x)$ .

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Schritt 3.4.4.1.1

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

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

Schritt 3.4.4.1.2

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

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

Schritt 3.4.4.1.3

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

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

Schritt 3.4.4.1.4

Addiere $1$ und $1$ .

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

Schritt 3.4.4.2

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

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Schritt 3.4.4.2.1

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

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

Schritt 3.4.4.2.2

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

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

Schritt 3.4.4.2.3

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

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

Schritt 3.4.4.3

Kombiniere Exponenten.

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Schritt 3.4.4.3.1

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

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

Schritt 3.4.4.3.2

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

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

Schritt 3.4.4.3.3

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

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

Schritt 3.4.4.3.4

Addiere $1$ und $1$ .

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

Schritt 3.4.5

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

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Schritt 3.4.5.1

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

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

Schritt 3.4.5.2

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

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

Schritt 3.4.5.3

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

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

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

Schritt 3.4.6

Kombiniere Exponenten.

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Schritt 3.4.6.1

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

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

Schritt 3.4.6.2

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

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

Schritt 3.4.6.3

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

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

Schritt 3.4.6.4

Addiere $1$ und $1$ .

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

Schritt 3.5

Vereinfache den Nenner.

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Schritt 3.5.1

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

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

Schritt 3.5.2

Da beide Terme perfekte Quadrate sind, faktorisiere durch Anwendung der dritten binomischen Formel, $a^{2} - b^{2} = (a + b) (a - b)$ , mit $a = \cos (x) \cdot 1$ und $b = \sin (x)$ .

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

Schritt 3.5.3

Vereinfache.

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Schritt 3.5.3.1

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

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

Schritt 3.5.3.2

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

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

Schritt 3.6

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

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Schritt 3.6.1

Kürze den gemeinsamen Faktor.

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

Schritt 3.6.2

Forme den Ausdruck um.

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

Schritt 3.7

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

$\cos^{2} (x)$

Schritt 4

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

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