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

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$L.H.S.$ $=\sin 3 x+\sin 2 x-\sin x$

$=\sin 3 x+\left[2 \cos \left(\frac{2 x+x}{2}\right) \sin \left(\frac{2 x-x}{2}\right)\right]$

$\left[\sin A-\sin B=2 \cos \left(\frac{A+B}{2}\right) \sin \left(\frac{A-B}{2}\right)\right]$

$=\sin 3 x+\left[2 \cos \left(\frac{3 x}{2}\right) \sin \left(\frac{x}{2}\right)\right]$

$=\sin 3 x+2 \cos \frac{3 x}{2} \sin \frac{x}{2}$

$=2 \sin \frac{3 x}{2} \cdot \cos \frac{3 x}{2}+2 \cos \frac{3 x}{2} \sin \frac{x}{2}$$\quad[\sin 2 A=2 \sin A \cdot \cos B]$

$=2 \cos \left(\frac{3 x}{2}\right)\left[\sin \left(\frac{3 x}{2}\right)+\sin \left(\frac{x}{2}\right)\right]$

$=2 \cos \left(\frac{3 x}{2}\right)\left[2 \sin \left\{\frac{\left(\frac{3 x}{2}\right)+\left(\frac{x}{2}\right)}{2}\right\} \cos \left\{\frac{\left(\frac{3 x}{2}\right)-\left(\frac{x}{2}\right)}{2}\right\}\right]$

$\left[\sin A+\sin B=2 \sin \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)\right]$

$=2 \cos \left(\frac{3 x}{2}\right) \cdot 2 \sin x \cos \left(\frac{x}{2}\right)$

$=4 \sin x \cos \left(\frac{x}{2}\right) \cos \left(\frac{3 x}{2}\right)= R. H.S.$

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