JEE MAIN - Mathematics (2022 - 25th July Evening Shift - No. 14)
$$2 \sin \left(\frac{\pi}{22}\right) \sin \left(\frac{3 \pi}{22}\right) \sin \left(\frac{5 \pi}{22}\right) \sin \left(\frac{7 \pi}{22}\right) \sin \left(\frac{9 \pi}{22}\right)$$ is equal to :
$$\frac{3}{16}$$
$$\frac{1}{16}$$
$$\frac{1}{32}$$
$$\frac{9}{32}$$
Explanation
$$2\sin {\pi \over {22}}\sin {{3\pi } \over {22}}\sin {{5\pi } \over {22}}\sin {{7\pi } \over {22}}\sin {{9\pi } \over {22}}$$
$$ = 2\sin \left( {{{11\pi - 10\pi } \over {22}}} \right)\sin \left( {{{11\pi - 8\pi } \over {22}}} \right)\sin \left( {{{11\pi - 6\pi } \over {22}}} \right)\sin \left( {{{11\pi - 4\pi } \over {22}}} \right)\sin \left( {{{11\pi - 2\pi } \over {22}}} \right)$$
$$ = 2\cos {\pi \over {11}}\cos {{2\pi } \over {11}}\cos {{3\pi } \over {11}}\cos {{4\pi } \over {11}}\cos {{5\pi } \over {11}}$$
$$ = {{2\sin {{32\pi } \over {11}}} \over {{2^5}\sin {\pi \over {11}}}}$$
$$ = {1 \over {16}}$$
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