Prove each of the following identities:

Consider L.H.S. =

Multiply and divide by (cosec θ + cot θ):

=

=

= (cosec θ + cot θ)²

Thus, proved.

Also, consider (cosec θ + cot θ)² = cosec² θ + cot² θ+ 2 cosec θ cot θ

= 1 + cot² θ + cot² θ+ 2 cosec θ cot θ (∵ 1 + cot² θ = cosec² θ)

= (1 + 2 cot² θ + 2 cosec θ cot θ)

= R.H.S.

Hence, proved.