Precalculus Examples

$\frac{\tan (θ)}{\sec (θ)} = \sin (θ)$

Step 1

Start on the left side.

$\frac{\tan (θ)}{\sec (θ)}$

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.

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

Step 2.2

Apply the reciprocal identity to $\sec (θ)$ .

$\frac{\frac{\sin (θ)}{\cos (θ)}}{\frac{1}{\cos (θ)}}$

Step 3

Simplify.

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{\sin (θ)}{\cos (θ)} \cos (θ)$

Step 3.2

Cancel the common factor of $\cos (θ)$ .

$\sin (θ)$

Step 4

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

$\frac{\tan (θ)}{\sec (θ)} = \sin (θ)$ is an identity