Prove the following identities.

Consider LHS,

LHS =

Multiplying and dividing by sinθ + cosθ + 1,

⇒ = ×

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

⇒ =

=

We know that 1 – cos²θ = sin²θ and sin²θ + cos²θ = 1.

=

=

=

=

We know that = secθ and = tanθ.

⇒ = secθ + tanθ

Multiplying and dividing by secθ – tanθ,

⇒ = secθ + tanθ ×

=

We know that sec² – tan²θ = 1.

∴ = = RHS

Hence proved.