Prove the following identities:

sin 4x = 4 sin x cos³ x – 4 cos x sin³ x

To prove: sin 4x = 4 sin x cos³ x – 4 cos x sin³ x

Proof:

Take LHS:

sin 4x

Identities used:

sin 2x = 2 sin x cos x

cos 2x = cos² x – sin² x

Therefore,

= 2 sin 2x cos 2x

= 2 (2 sin x cos x) (cos² x – sin² x)

= 4 sin x cos x (cos² x – sin² x)

= 4 sin x cos³ x – 4 sin³ x cos x

= RHS

Hence Proved