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

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Solution

$L . H . S = (sin 3 x + sin x) sin x + (cos 3 x - cos x) cos x$
$= [2 sin (\frac{3 x + x}{2}) cos (\frac{3 x - x}{2})] sin x + [- 2 sin (\frac{3 x + x}{2}) sin (\frac{3 x - x}{2})] cos x$
$= [2 sin 2 x cos x] sin x + [- 2 sin 2 x sin x] cos x$
$= sin 2 x (2 sin x cos x) - sin 2 x (2 sin x cos x)$
$= sin 2 x \cdot sin 2 x - sin 2 x \cdot sin 2 x$
$= 0 = R . H . S$

Hence proved.

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