# 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

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