Trigonometry Examples

$\frac{\tan (x) \cot (x)}{\cos (x)} = \sec (x)$

Step 1

Start on the left side.

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

Step 2

Convert to sines and cosines.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

Simplify.

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Step 3.1

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

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

Step 3.2

Simplify the numerator.

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

Step 4

Rewrite $\frac{1}{\cos (x)}$ as $\sec (x)$ .

$\sec (x)$

Step 5

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

$\frac{\tan (x) \cot (x)}{\cos (x)} = \sec (x)$ is an identity