Prove that : sin 3x + sin 2x - sin x = 4 ;x x 2 3x 2 - Vidyadip

Question

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

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Answer

We have L.H.S. = sin 3x + sin 2x - sinx
$=\left[2 \cos \left(\frac{3 x+x}{2}\right) \sin \left(\frac{3 x-x}{2}\right)\right]+2 \sin x \cos x$
$\left[\because \sin C-\sin D=2 \cos \frac{C+D}{2} \cdot \sin \frac{C-D}{2}\right]$
$=2 \cos 2 x \sin x+2 \sin x \cos x$
$=2 \sin x[\cos 2 x+\cos x]$
$=2 \sin x\left[2 \cos \left(\frac{2 x+x}{2}\right) \cos \left(\frac{2 x-x}{2}\right)\right] \quad=2 \sin x\left[2 \cos \left(\frac{3 x}{2}\right) \cos \left(\frac{x}{2}\right)\right]$
$=4 \sin x \cos \frac{x}{2} \cos \frac{3 x}{2}=R . H . S$

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