Trigonometrie Beispiele

\frac{{csc}^{2} (x)}{{cot}^{2} (x) - 1} = \frac{{sec}^{2} (x)}{1 - {tan}^{2} (x)}

$\frac{\csc^{2} (x)}{\cot^{2} (x) - 1} = \frac{\sec^{2} (x)}{1 - \tan^{2} (x)}$

Schritt 1

Beginne auf der linken Seite.

\frac{{csc}^{2} (x)}{{cot}^{2} (x) - 1}

$\frac{\csc^{2} (x)}{\cot^{2} (x) - 1}$

Schritt 2

Wandle in Sinus und Kosinus um.

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Schritt 2.1

Wende die Kehrwertfunktion auf

csc (x)

$\csc (x)$ an.

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

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

Schritt 2.2

Schreibe

cot (x)

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

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

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

Schritt 2.3

Wende die Produktregel auf

\frac{1}{sin (x)}

$\frac{1}{\sin (x)}$ an.

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

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

Schritt 2.4

Wende die Produktregel auf

\frac{cos (x)}{sin (x)}

$\frac{\cos (x)}{\sin (x)}$ an.

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

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

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

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

Schritt 3

Vereinfache.

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Schritt 3.1

Multipliziere den Zähler mit dem Kehrwert des Nenners.

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

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

Schritt 3.2

Eins zu einer beliebigen Potenz erhoben ergibt eins.

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

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

Schritt 3.3

Vereinfache den Nenner.

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Schritt 3.3.1

Schreibe

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

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

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

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

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

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

Schritt 3.3.2

Schreibe

1

$1$ als

1^{2}

$1^{2}$ um.

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

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

Schritt 3.3.3

Da beide Terme perfekte Quadrate sind, faktorisiere durch Anwendung der dritten binomischen Formel,

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ , mit

a = \frac{cos (x)}{sin (x)}

$a = \frac{\cos (x)}{\sin (x)}$ und

b = 1

$b = 1$ .

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

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

Schritt 3.3.4

Schreibe

1

$1$ als Bruch mit einem gemeinsamen Nenner.

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

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

Schritt 3.3.5

Vereinige die Zähler über dem gemeinsamen Nenner.

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

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

Schritt 3.3.6

- 1

$- 1$ als Bruch mit einem gemeinsamen Nenner zu schreiben, multipliziere mit

\frac{sin (x)}{sin (x)}

$\frac{\sin (x)}{\sin (x)}$ .

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

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

Schritt 3.3.7

Kombiniere

- 1

$- 1$ und

\frac{sin (x)}{sin (x)}

$\frac{\sin (x)}{\sin (x)}$ .

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

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

Schritt 3.3.8

Vereinige die Zähler über dem gemeinsamen Nenner.

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

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

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

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

Schritt 3.4

Mutltipliziere

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

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

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

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

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

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

Schritt 3.5

Vereinfache den Nenner.

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Schritt 3.5.1

Potenziere

sin (x)

$\sin (x)$ mit

1

$1$ .

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

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

Schritt 3.5.2

Potenziere

$\sin (x)$ mit

$1$ .

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

Schritt 3.5.3

Wende die Exponentenregel

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

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

Schritt 3.5.4

Addiere

$1$ und

$1$ .

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

Schritt 3.6

Kombinieren.

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

Schritt 3.7

Mutltipliziere

$1$ mit

$1$ .

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

Schritt 3.8

Kombiniere

${\sin (x)}^{2}$ und

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

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

Schritt 3.9

Vereinfache den Ausdruck durch Kürzen der gemeinsamen Faktoren.

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

Schritt 4

Schreibe

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

$\frac{\sec^{2} (x)}{1 - \tan^{2} (x)}$ um.

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

Schritt 5

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

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