How do I prove that sin theta + sin theta *cot squared theta= csc theta

Prove:
sin(\theta) +sin(\theta)cot^2(\theta)=csc(\theta)

Use these three identities. The first two are definitions of the cosecant and cotangent, respectively.

csc(\theta) = \frac {1}{sin(\theta)}

cot(\theta) = \frac {cos(\theta)}{sin(\theta)}

cos^2(\theta) + sin^2(\theta) = 1

Use the cot and csc to get everything in terms of sines and cosines. Manipulate the result algebraically. The third identity can be used to eliminate cosines. You'll end up with the same thing on the left and right of the equals sign (quod erat demonstrandum).