Question

$\sin \frac{\pi }{18}\sin \frac{3\pi }{18}\sin \frac{5\pi }{18}\sin \frac{7\pi }{18}=\frac{1}{16}$

Answer 1

$\sin {10}^o$$\sin {30}^o$$\sin {50}^o$$\sin {70}^o$

$=\sin {30}^o$$(\sin {10}^o$$\sin {50}^o)$$\sin {70}^o$

$=\frac{1}{2}$$(\sin {10}^o$$\sin {50}^o)$$\sin {70}^o$

$=\frac{1}{2}\times$$\frac{1}{2}(2\times \sin {10}^o$$\sin {50}^o)$$\sin {70}^o$

$=\frac{1}{4}\left(\cos \right({50}^o-{10}^o)-\cos ({50}^o+{10}^o\left)\right)\sin {70}^o$

$=\frac{1}{4}(cos{40}^o-cos{60}^o)sin{70}^o$

$=\frac{1}{4}(\sin {70}^o\cos {40}^o-\frac{1}{2}sin{70}^o)$

$=\frac{1}{4}(\frac{1}{2}$$\left(\sin \right({70}^o+{40}^o)$+$\sin ({70}^o-{40}^o)$) $-\frac{1}{8}\sin {70}^o$

$=\frac{1}{8}$ $\sin {110}^o+$$\frac{1}{8}\sin {30}^o-$$\frac{1}{8}\sin {70}^o$

$=\frac{1}{16}$