When csc(theta) sin(theta) is simplified, what is the result?

Cosec theta = 1/sin theta 

Therefore, csc theta * sin theta = 1. 

cosec theta = 1/sin theta, sec theta = 1 / cos theta. 

Therefore, Equation 2 reduces to, 
(1/sin theta) / (1/cos theta) = cos theta/sin theta = 1/tan theta = cot theta. 

tan theta + cot theta = tan theta + 1/tan theta. Taking LCM, 
= (1 + tan^2 theta)/tan theta 
From the identities, 1 + tan^2 theta = sec^2 theta. 
Hence, equation 3 reduces to, 
sec^2 theta/tan theta = (1/cos^2 theta) / (sin theta/cos theta) 
= cos theta / (sin theta * cos^2 theta) 
= 1 / (sin theta*cos theta). This cannot be reduced, hence is equal to sec theta*cosec theta. 

1-sin^2 theta / 1+sin theta = (1 + sin theta)*(1-sin theta) / (1 + sin theta) 
= 1 - sin theta. 

Hope this helped :)