Trigonometry 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 $\cot (- x)$ is an odd function, rewrite $\cot (- x)$ as $- \cot (x)$ .

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

Step 7

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

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Step 7.1

Add parentheses.

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

Step 7.2

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

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

Step 7.3

Cancel the common factors.

$- \cos (x)$