Determine if True cos(theta)cot(theta)=csc(theta)-sin(theta)

$\cos (θ) \cot (θ) = \csc (θ) - \sin (θ)$

Step 1

Start on the right side.

$\csc (θ) - \sin (θ)$

Step 2

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

$\frac{1}{\sin (θ)} - \sin (θ)$

Step 3

Write $- \sin (θ)$ as a fraction with denominator $1$ .

$\frac{1}{\sin (θ)} + \frac{- \sin (θ)}{1}$

Step 4

Add fractions.

To write $\frac{- \sin (θ)}{1}$ as a fraction with a common denominator, multiply by $\frac{\sin (θ)}{\sin (θ)}$ .

$\frac{1}{\sin (θ)} + \frac{- \sin (θ)}{1} \cdot \frac{\sin (θ)}{\sin (θ)}$

Multiply $\frac{- \sin (θ)}{1}$ by $\frac{\sin (θ)}{\sin (θ)}$ .

$\frac{1}{\sin (θ)} + \frac{- \sin (θ) \sin (θ)}{\sin (θ)}$

Combine the numerators over the common denominator.

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

Step 5

Multiply $- \sin (θ) \sin (θ)$ .

$\frac{1 - \sin^{2} (θ)}{\sin (θ)}$

Step 6

Apply pythagorean identity.

$\frac{\cos^{2} (θ)}{\sin (θ)}$

Step 7

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

$\cos (θ) \cot (θ)$

Step 8

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

$\cos (θ) \cot (θ) = \csc (θ) - \sin (θ)$ is an identity