Precalculus Examples

$\frac{\cot (x)}{\csc (- x)}$

Step 1

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

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

Step 2

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

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

Step 3

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\sin (- x)}$ .

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

Step 4

Write $\sin (- x)$ as a fraction with denominator $1$ .

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

Step 5

Simplify.

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Step 5.1

Divide $\sin (- x)$ by $1$ .

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

Step 5.2

Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

$\cot (x) \sin (- x)$

Step 6

Since $\sin (- x)$ is an odd function, rewrite $\sin (- x)$ as $- \sin (x)$ .

$\cot (x) (- \sin (x))$

Step 7

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 7.1

Reorder $\cot (x)$ and $- \sin (x)$ .

$- \sin (x) \cot (x)$

Step 7.2

Add parentheses.

$- (\sin (x) \cot (x))$

Step 7.3

Reorder $\sin (x)$ and $\cot (x)$ .

$- (\cot (x) \sin (x))$

Step 7.4

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

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

Step 7.5

Cancel the common factors.

$- \cos (x)$