Proof:

Take LHS:

Identities used:

(a + b)² = a² + b² + 2ab

a³ – b³ = (a – b) (a² + b² + ab)

Therefore,

{∵ cos 2x = cos² x – sin² x}

{∵ sin² x + cos² x = 1}

{∵ sin 2x = 2 sin x cos x}

= RHS

Hence Proved