Question

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

Answer 1

(a) $\sin 17\mathrm{\theta }-\sin 11\mathrm{\theta }$

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