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  4. 2 sin ((π/22)) sin ((3 π/22)) sin ((5 π/22)) sin ((7 π/22)) sin ((9 π/22)) is equal to

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Q. $2 \sin \left(\frac{\pi}{22}\right) \sin \left(\frac{3 \pi}{22}\right) \sin \left(\frac{5 \pi}{22}\right) \sin \left(\frac{7 \pi}{22}\right) \sin \left(\frac{9 \pi}{22}\right)$ is equal to

JEE MainJEE Main 2022Trigonometric Functions

Solution:

$2 \sin \left(\frac{\pi}{22}\right) \sin \left(\frac{3 \pi}{22}\right) \sin \left(\frac{5 \pi}{22}\right) \sin \left(\frac{7 \pi}{22}\right) \sin \left(\frac{9 \pi}{22}\right)$
$2 \cos \left(\frac{\pi}{2}-\frac{\pi}{22}\right) \cos \left(\frac{\pi}{2}-\frac{3 \pi}{22}\right) \cos \left(\frac{\pi}{2}-\frac{5 \pi}{22}\right) \cos \left(\frac{\pi}{2}-\frac{7 \pi}{15}\right) $
$ \cos \left(\frac{\pi}{2}-\frac{9 \pi}{22}\right)$
$2 \cos \left(\frac{10 \pi}{22}\right) \cos \left(\frac{8 \pi}{22}\right) \cos \left(\frac{6 \pi}{22}\right) \cos \left(\frac{4 \pi}{22}\right) \cos \left(\frac{2 \pi}{22}\right)$
$ 2 \cos \left(\frac{\pi}{11}\right) \cos \left(\frac{2 \pi}{11}\right) \cos \left(\frac{3 \pi}{11}\right) \cos \left(\frac{4 \pi}{11}\right) \cos \left(\frac{5 \pi}{11}\right)$
$ 2 \cos \left(\frac{\pi}{11}\right) \cos \left(\frac{2 \pi}{11}\right) \cos \left(\frac{4 \pi}{11}\right) \cos \left(\pi-\frac{3 \pi}{11}\right) \cos \left(\pi+\frac{5 \pi}{11}\right) $
$2 \cos \left(\frac{\pi}{11}\right) \cos \left(\frac{2 \pi}{11}\right) \cos \left(\frac{4 \pi}{11}\right) \cos \left(\frac{8 \pi}{11}\right) \cos \left(\frac{16 \pi}{11}\right)$
$ \frac{2 \cdot \sin \left(2^5 \times \frac{\pi}{11}\right)}{2^5 \sin \frac{\pi}{11}} $
$ \frac{2 \cdot \sin \left(\frac{32 \pi}{11}\right)}{32 \sin \frac{\pi}{11}}=\frac{1}{16}$