Prove the following identities

Consider LHS,

LHS =

Multiplying and dividing with (cosecθ - cotθ),

⇒ = ×

We know that (a + b) (a – b) = a² – b².

⇒ =

We know that cosec²θ - cot²θ = 1.

⇒ =

We know that = cosecθ and = cotθ.

⇒ = sinθ () - sinθ ()

= 1 – cosθ = RHS

Hence proved.