Prove that : (sin 3x + sin x) sin x + (cos 3x - cos x) cos x = 0 - Vidyadip

Question

Prove that : (sin 3x + sin x) sin x + (cos 3x - cos x) cos x = 0

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Answer

We have L.H.S. = (sin 3x + sin x) sin x + (cos 3x - cosx) cos x
$ = \left[ {2\sin \left( {\frac{{3x + x}}{2}} \right)\cos \left( {\frac{{3x - x}}{2}} \right)} \right]\sin x$$ + \left[ { - 2\sin \left( {\frac{{3x + x}}{2}} \right)\sin \left( {\frac{{3x - x}}{2}} \right)} \right]\cos x$
= [2sin 2xcosx]sinx + [-2sin 2xsin x] cosx
= 2sin 2xcosxsin x - 2 sin 2x cos x sin x
= 0 = R.H.S.

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