Trigonometry Examples

$\frac{\tan (θ)}{\cot (θ)} = \tan^{2} (θ)$

Step 1

Start on the left side.

$\frac{\tan (θ)}{\cot (θ)}$

Step 2

Convert to sines and cosines.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

Simplify.

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.2

Multiply $\frac{\sin (θ)}{\cos (θ)} \cdot \frac{\sin (θ)}{\cos (θ)}$ .

$\frac{\sin^{2} (θ)}{\cos^{2} (θ)}$

Step 4

Rewrite $\frac{\sin^{2} (θ)}{\cos^{2} (θ)}$ as $\tan^{2} (θ)$ .

$\tan^{2} (θ)$

Step 5

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

$\frac{\tan (θ)}{\cot (θ)} = \tan^{2} (θ)$ is an identity