Precalculus Examples

$\frac{\cot (x) \tan (x)}{\sin (x)} = \csc (x)$

Step 1

Start on the left side.

$\frac{\cot (x) \tan (x)}{\sin (x)}$

Step 2

Convert to sines and cosines.

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Step 2.1

Write $\cot (x)$ in sines and cosines using the quotient identity.

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

Step 2.2

Write $\tan (x)$ in sines and cosines using the quotient identity.

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

Step 3

Simplify.

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Step 3.1

Multiply $\frac{\cos (x)}{\sin (x)}$ by $\frac{\sin (x)}{\cos (x)}$ .

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

Step 3.2

Simplify the numerator.

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

Step 4

Rewrite $\frac{1}{\sin (x)}$ as $\csc (x)$ .

$\csc (x)$

Step 5

Because the two sides have been shown to be equivalent, the equation is an identity.

$\frac{\cot (x) \tan (x)}{\sin (x)} = \csc (x)$ is an identity