Determine if True (tan(theta))/(sin(theta))=sec(theta)

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

Start on the left side.

$\frac{\tan (θ)}{\sin (θ)}$

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

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

Simplify.

Multiply the numerator by the reciprocal of the denominator.

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

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

$\frac{1}{\cos (θ)}$

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

$\sec (θ)$

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

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