Question

$2\left(1-2{\sin }^{2}7x\right)\sin 3x$ is equal to
(a) $\sin 17x-\sin 11x$
(b) $\sin 11x-\sin 17x$
(c) $\cos 17x-\cos 11x$
(d) $\cos 17x+\cos 11x$

Answer 1

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

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ 2\left(1-2{\sin }^{2}7x\right)\sin 3x=2\left(\cos 14x\right)\sin 3x \\ \left[\because \cos 2x=1-2{\sin }^{2}x\right] \\ =2\sin 3x\cos 14x \\ =\sin 17x-\sin 11x \\ \left[\because 2\mathrm{sinA}\mathrm{cosB}=\sin \left(\mathrm{A}+\mathrm{B}\right)-\sin \left(\mathrm{A}-\mathrm{B}\right)\right] \\ \therefore 2\left(1-2{\sin }^{2}7x\right)\sin 3x=\sin 17x-\sin 11x \end{gathered}$$