Prove the following identities

Consider LHS,

LHS =

Multiplying and dividing with (secθ + tanθ),

⇒ = ×

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

⇒ =

We know that sec²θ - tan²θ = 1.

⇒ =

We know that = secθ and = tanθ.

⇒ = cosθ() + cosθ ()

= 1 + sinθ = RHS

Hence proved.