Question:

$2\left(1-2 \sin ^{2} 7 x\right) \sin 3 x$ is equal to

(a) $\sin 17 x-\sin 11 x$

(b) $\sin 11 x-\sin 17 x$

(c) $\cos 17 x-\cos 11 x$

(d) $\cos 17 x+\cos 11 x$

Solution:

(a) $\sin 17 x-\sin 11 x$

We have,

$2\left(1-2 \sin ^{2} 7 x\right) \sin 3 x=2(\cos 14 x) \sin 3 x$

$\left[\because \cos 2 x=1-2 \sin ^{2} x\right]$

$=2 \sin 3 x \cos 14 x$

$=\sin 17 x-\sin 11 x$

$[\because 2 \sin \mathrm{A} \cos \mathrm{B}=\sin (\mathrm{A}+\mathrm{B})-\sin (\mathrm{A}-\mathrm{B})]$

$\therefore 2\left(1-2 \sin ^{2} 7 x\right) \sin 3 x=\sin 17 x-\sin 11 x$