prove that [csc^2theta-cot theta]^2=1-cos/1+cos theta

Question

give its solution

Answer 1

The identity has been wrongly stated. It should be:

(cosecθ-cotθ)²=(1-cosθ)/(1+cosθ).

Take the right-hand side expression and multiply top and bottom by 1-cosθ:

(1-cosθ)²/(1-cos²θ)=(1-2cosθ+cos²θ)/sin²θ=

cosec²θ-2cotθcosecθ+cot²θ=(cosecθ-cotθ)², which is the left-hand side.

This proves (cosecθ-cotθ)²=(1-cosθ)/(1+cosθ) QED