Prove the following identities:

Proof:

Take LHS:

Identities used:

cos 2x = cos² x – sin² x

sin 2x = 2 sin x cos x

Therefore,

{∵ a² – b² = (a - b)(a + b) & sin² x + cos² x = 1}

{∵ a² + b² + 2ab = (a + b)²}

Multiplying numerator and denominator by

{∵ sin (A – B) = sin A cos B – sin B cos A

cos (A – B) = cos A cos B + sin A sin B}

= RHS

Hence Proved