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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