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Question

Prove that:
$s i n 3 x + s i n 2 x - s i n x = 4 s i n x c o s \frac{x}{2} c o s \frac{3 x}{2}$

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Solution

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$L H S = s i n 3 x + s i n 2 x - s i n x$
$= 2 sin (\frac{3 x + 2 x}{2}) cos (\frac{3 x - 2 x}{2}) - 2 sin \frac{x}{2} cos \frac{x}{2}$
$= 2 cos \frac{x}{2} (sin \frac{5 x}{2} - sin \frac{x}{2}) = 2 cos \frac{x}{2} cos \frac{3 x}{2} sin x$
$= 4 s i n x c o s \frac{x}{2} c o s \frac{3 x}{2} = R H S$

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