Prove the following identities.

Consider LHS,

LHS = -

We know that cosec²θ – cot²θ = 1 and = cosecθ.

⇒ - = – cosecθ

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

⇒ - = – cosecθ

= cosecθ + cotθ – cosecθ

= cotθ … (1)

Now consider RHS,

RHS = -

We know that cosec²θ – cot²θ = 1 and = cosecθ.

⇒ - = cosecθ -

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

⇒ - =cosecθ -

= cosecθ – (cosecθ – cotθ)

= cotθ … (2)

From (1) and (2), LHS = RHS

Hence proved.