Determine if True (cot(theta))/(csc(theta))=cos(theta)

$\frac{\cot (θ)}{\csc (θ)} = \cos (θ)$

Start on the left side.

$\frac{\cot (θ)}{\csc (θ)}$

Convert to sines and cosines.

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

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

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

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

Simplify.

Multiply the numerator by the reciprocal of the denominator.

$\frac{\cos (θ)}{\sin (θ)} \sin (θ)$

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

$\cos (θ)$

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

$\frac{\cot (θ)}{\csc (θ)} = \cos (θ)$ is an identity