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Trigonometric Ratios of Compound Angles

Prove: sin3x...

Question

Prove:
$sin 3 x + sin 2 x - sin x = 4 sin x cos \frac{x}{2} cos \frac{3 x}{2}$

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Solution

$sin 3 x + sin 2 x - sin x$
$= sin 3 x + 2 cos \frac{3 x}{2} sin \frac{x}{2}$
$= 2 sin \frac{3 x}{2} cos \frac{3 x}{2} + 2 cos \frac{3 x}{2} sin \frac{x}{2}$
$= 2 cos \frac{3 x}{2} (sin \frac{3 x}{2} + sin \frac{x}{2})$
$= (2 cos \frac{3 x}{2}) (2 sin x cos \frac{x}{2})$
$= 4 sin x cos \frac{x}{2} cos \frac{3 x}{2}$
Here proved

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