Prove the following identities.

Consider LHS,

LHS =

We know that cot²θ = cosec² – 1 and tan²θ = sec²θ – 1.

⇒ =

=

= 1 … (1)

Consider RHS,

RHS = (sinθcosθ) (tanθ + cotθ)

Expanding,

⇒ (sinθcosθ) (tanθ + cotθ) = sinθcosθtanθ + sinθcosθcotθ

We know that = tanθ and = cotθ.

⇒ (sinθcosθ) (tanθ + cotθ) = sinθcosθ() + sinθcosθ()

= sin²θ + cos²θ

We know that sin²θ + cos²θ = 1.

∴ (sinθcosθ) (tanθ + cotθ) = 1 … (2)

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

Hence proved.