3. trignometrical ratios,functions and identities — ગણિત ધોરણ 11 સાયન્સ — Question

$ = \frac{1}{4}\left[ {2\sin \frac{\pi }{{16}}\sin \frac{{3\pi }}{{16}}.2\sin \frac{{5\pi }}{{16}}\sin \frac{{7\pi }}{{16}}} \right]$

$ = \frac{1}{4}\left[ {\left( {\cos \frac{\pi }{8} - \cos \frac{\pi }{4}} \right)\left( {\cos \frac{\pi }{8} - \cos \frac{{3\pi }}{4}} \right)} \right]$

$ = \frac{1}{4}\left[ {\left( {\cos \frac{\pi }{8} - \frac{1}{{\sqrt 2 }}} \right)\left( {\cos \frac{\pi }{8} + \frac{1}{{\sqrt 2 }}} \right)} \right]$

$ = \frac{1}{4}\left[ {\left( {{{\cos }^2}\frac{\pi }{8} - \frac{1}{2}} \right)} \right] $

$= \frac{1}{8}\left[ {2{{\cos }^2}\frac{\pi }{8} - 1} \right]$

$ = \frac{1}{8}\left[ {\cos \frac{\pi }{4}} \right] $

$= \frac{1}{8} \times \frac{1}{{\sqrt 2 }} $

$= \frac{{\sqrt 2 }}{{16}}$.