Algebra Beispiele

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

Schritt 1

Vereinfache den Zähler.

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Schritt 1.1

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

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

Schritt 1.2

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

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

Schritt 2

Vereinfache den Nenner.

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Schritt 2.1

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

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

Schritt 2.2

Vereinfache.

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Schritt 2.2.1

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

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

Schritt 2.2.2

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

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Schritt 2.2.2.1

Faktorisiere $\cos (x)$ aus $\frac{\cos (x)}{\sin (x)}$ heraus.

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

Schritt 2.2.2.2

Multipliziere mit $1$ .

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

Schritt 2.2.2.3

Faktorisiere $\cos (x)$ aus $\cos (x) \frac{1}{\sin (x)} + \cos (x) \cdot 1$ heraus.

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

Schritt 2.2.3

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

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

Schritt 2.2.4

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

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Schritt 2.2.4.1

Faktorisiere $\cos (x)$ aus $\frac{\cos (x)}{\sin (x)}$ heraus.

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

Schritt 2.2.4.2

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

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

Schritt 2.2.4.3

Faktorisiere $\cos (x)$ aus $\cos (x) \frac{1}{\sin (x)} + \cos (x) \cdot - 1$ heraus.

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

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

Schritt 2.2.5

Kombiniere Exponenten.

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Schritt 2.2.5.1

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

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

Schritt 2.2.5.2

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

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

Schritt 2.2.5.3

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

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

Schritt 2.2.5.4

Addiere $1$ und $1$ .

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

Schritt 3

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

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

Schritt 4

Multipliziere $\cos^{2} (x)$ mit $\cos^{2} (x)$ durch Addieren der Exponenten.

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Schritt 4.1

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

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

Schritt 4.2

Addiere $2$ und $2$ .

$\frac{\frac{\cos^{4} (x)}{\sin^{2} (x)}}{\cos^{2} (x) (\frac{1}{\sin (x)} + 1) (\frac{1}{\sin (x)} - 1)}$

Schritt 5

Multipliziere den Zähler mit dem Kehrwert des Nenners.

$\frac{\cos^{4} (x)}{\sin^{2} (x)} \cdot \frac{1}{\cos^{2} (x) (\frac{1}{\sin (x)} + 1) (\frac{1}{\sin (x)} - 1)}$

Schritt 6

Kombinieren.

$\frac{\cos^{4} (x) \cdot 1}{\sin^{2} (x) (\cos^{2} (x) (\frac{1}{\sin (x)} + 1) (\frac{1}{\sin (x)} - 1))}$

Schritt 7

Kürze den gemeinsamen Teiler von $\cos^{4} (x)$ und $\cos^{2} (x)$ .

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Schritt 7.1

Faktorisiere $\cos^{2} (x)$ aus $\cos^{4} (x) \cdot 1$ heraus.

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

Schritt 7.2

Kürze die gemeinsamen Faktoren.

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Schritt 7.2.1

Faktorisiere $\cos^{2} (x)$ aus $\sin^{2} (x) (\cos^{2} (x) (\frac{1}{\sin (x)} + 1) (\frac{1}{\sin (x)} - 1))$ heraus.

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

Schritt 7.2.2

Kürze den gemeinsamen Faktor.

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

Schritt 7.2.3

Forme den Ausdruck um.

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

Schritt 8

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

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

Schritt 9

Multipliziere mit $1$ .

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

Schritt 10

Multipliziere mit $1$ .

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

Schritt 11

Separiere Brüche.

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

Schritt 12

Wandle von $\frac{\cos^{2} (x)}{\sin^{2} (x)}$ nach $\cot^{2} (x)$ um.

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

Schritt 13

Mutltipliziere $\frac{1}{\sin (x)} + 1$ mit $1$ .

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

Schritt 14

Vereinfache den Nenner.

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Schritt 14.1

Wandle von $\frac{1}{\sin (x)}$ nach $\csc (x)$ um.

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

Schritt 14.2

Wandle von $\frac{1}{\sin (x)}$ nach $\csc (x)$ um.

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

Schritt 15

Kombiniere $\frac{1}{(\csc (x) + 1) (\csc (x) - 1)}$ und $\cot^{2} (x)$ .

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