Q. 15

# Prove the following identities:(sin 3x + sin x) sin x + (cos 3x – cos x) cos x = 0

To prove: (sin 3x + sin x)sin x + (cos 3x – cos x)cos x= 0

Proof:

Take LHS:

(sin 3x + sin x)sin x + (cos 3x – cos x)cos x

= (sin 3x)(sin x) + sin2 x + (cos 3x)(cos x) – cos2 x

= [(sin 3x)(sin x) + (cos 3x)(cos x)] + (sin2 x – cos2 x)

= [(sin 3x)(sin x) + (cos 3x)(cos x)] – (cos2 x – sin2 x)

= cos(3x – x) – cos 2x

{ cos 2x = cos2 x – sin2 x &

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

= cos 2x – cos 2x

= 0

= RHS

Hence Proved

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