Calculus Examples

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

Step 1

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

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Step 1.1

Rewrite $\cot (x)$ in terms of sines and cosines.

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

Step 1.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.3

Factor $\cos (x)$ out of $\cos (x) + \frac{\cos (x)}{\sin (x)}$ .

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Step 1.3.1

Multiply by $1$ .

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

Step 1.3.2

Factor $\cos (x)$ out of $\frac{\cos (x)}{\sin (x)}$ .

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

Step 1.3.3

Factor $\cos (x)$ out of $\cos (x) \cdot 1 + \cos (x) \frac{1}{\sin (x)}$ .

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

Step 1.4

Separate fractions.

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

Step 1.5

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

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

Step 1.6

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

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

Step 1.7

Combine $\frac{1}{1 + \csc (x)}$ and $\sec (x)$ .

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