• # question_answer The value of$\sin \frac{\pi }{14}\sin \frac{3\pi }{14}\sin \frac{5\pi }{14}\sin \frac{7\pi }{14}\sin \frac{9\pi }{14}\sin \frac{11\pi }{14}\sin \frac{13\pi }{14}$ is equal to [IIT 1991; MNR 1992] A) $\frac{1}{8}$ B) $\frac{1}{16}$ C) $\frac{1}{32}$ D) $\frac{1}{64}$

$\sin \frac{\pi }{14}\sin \frac{3\pi }{14}\sin \frac{5\pi }{14}\sin \frac{7\pi }{14}\sin \frac{9\pi }{14}\sin \frac{11\pi }{14}\sin \frac{13\pi }{14}$ $=\sin \frac{\pi }{14}\sin \frac{3\pi }{14}\sin \frac{5\pi }{14}\times 1$ $\times \sin \left( \pi -\frac{5\pi }{14} \right)\sin \left( \pi -\frac{3\pi }{14} \right)\sin \left( \pi -\frac{\pi }{14} \right)$ $={{\left[ \sin \frac{\pi }{14}\sin \frac{3\pi }{14}\sin \frac{5\pi }{14}\sin \frac{7\pi }{14} \right]}^{2}}=\frac{1}{64}$.