Analysis Beispiele

r = {(\cos (x) + \cot (x))}^{- 1}

$r = {(\cos (x) + \cot (x))}^{- 1}$

Schritt 1

Vereinfache

{(\cos (x) + \cot (x))}^{- 1}

${(\cos (x) + \cot (x))}^{- 1}$ .

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Schritt 1.1

Schreibe

\cot (x)

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

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

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

Schritt 1.2

Schreibe den Ausdruck um mithilfe der Regel des negativen Exponenten

b^{- n} = \frac{1}{b^{n}}

$b^{- n} = \frac{1}{b^{n}}$ .

r = \frac{1}{\cos (x) + \frac{\cos (x)}{\sin (x)}}

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

Schritt 1.3

Faktorisiere

\cos (x)

$\cos (x)$ aus

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

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

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Schritt 1.3.1

Multipliziere mit

1

$1$ .

r = \frac{1}{\cos (x) \cdot 1 + \frac{\cos (x)}{\sin (x)}}

$r = \frac{1}{\cos (x) \cdot 1 + \frac{\cos (x)}{\sin (x)}}$

Schritt 1.3.2

Faktorisiere

\cos (x)

$\cos (x)$ aus

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

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

r = \frac{1}{\cos (x) \cdot 1 + \cos (x) \frac{1}{\sin (x)}}

$r = \frac{1}{\cos (x) \cdot 1 + \cos (x) \frac{1}{\sin (x)}}$

Schritt 1.3.3

Faktorisiere

\cos (x)

$\cos (x)$ aus

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

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

r = \frac{1}{\cos (x) (1 + \frac{1}{\sin (x)})}

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

r = \frac{1}{\cos (x) (1 + \frac{1}{\sin (x)})}

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

Schritt 1.4

Separiere Brüche.

r = \frac{1}{1 + \frac{1}{\sin (x)}} \cdot \frac{1}{\cos (x)}

$r = \frac{1}{1 + \frac{1}{\sin (x)}} \cdot \frac{1}{\cos (x)}$

Schritt 1.5

Wandle von

\frac{1}{\cos (x)}

$\frac{1}{\cos (x)}$ nach

\sec (x)

$\sec (x)$ um.

r = \frac{1}{1 + \frac{1}{\sin (x)}} \sec (x)

$r = \frac{1}{1 + \frac{1}{\sin (x)}} \sec (x)$

Schritt 1.6

Wandle von

\frac{1}{\sin (x)}

$\frac{1}{\sin (x)}$ nach

\csc (x)

$\csc (x)$ um.

r = \frac{1}{1 + \csc (x)} \sec (x)

$r = \frac{1}{1 + \csc (x)} \sec (x)$

Schritt 1.7

Kombiniere

\frac{1}{1 + \csc (x)}

$\frac{1}{1 + \csc (x)}$ und

\sec (x)

$\sec (x)$ .

r = \frac{\sec (x)}{1 + \csc (x)}

$r = \frac{\sec (x)}{1 + \csc (x)}$

r = \frac{\sec (x)}{1 + \csc (x)}

$r = \frac{\sec (x)}{1 + \csc (x)}$