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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
$\frac{1}{8}$
$\frac{1}{{16}}$
$\frac{1}{{32}}$
$\frac{1}{{64}}$
Solution
(d) $\sin \frac{\pi }{{14}}\sin \frac{{3\pi }}{{14}}\sin \frac{{5\pi }}{{14}}\sin \frac{{7\pi }}{{14}}\sin \frac{{9\pi }}{{14}}\sin \frac{{11\pi }}{{14}}\sin \frac{{13\pi }}{{14}}$
$ = \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}}$.
