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  4. The value of cos (π /11)cos ⁡ (2 π /11)cos ⁡ (3 π /11) ldots ldots cos ⁡ (11 π /11) is equal to

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Q. The value of $cos \frac{\pi }{11}cos ⁡ \frac{2 \pi }{11}cos ⁡ \frac{3 \pi }{11}\ldots \ldots cos ⁡ \frac{11 \pi }{11}$ is equal to

NTA AbhyasNTA Abhyas 2020

Solution:

Required expression $=\left(cos \frac{\pi }{11} cos ⁡ \frac{2 \pi }{11} cos ⁡ \frac{3 \pi }{11} cos ⁡ \frac{4 \pi }{11} cos ⁡ \frac{5 \pi }{11}\right)^{2}$
$=\left(cos \frac{\pi }{11} cos ⁡ \frac{2 \pi }{11} cos ⁡ \frac{8 \pi }{11} cos ⁡ \frac{4 \pi }{11} cos ⁡ \frac{5 \pi }{11}\right)^{2}$
$=\left(\frac{sin \frac{16 \pi }{11}}{16 sin ⁡ \frac{\pi }{11}} \times cos ⁡ \frac{5 \pi }{11}\right)^{2}$
$=\left(\frac{2 sin \frac{5 \pi }{11} cos ⁡ \frac{5 \pi }{11}}{32 sin ⁡ \frac{\pi }{11}}\right)^{2}$
$=\left(\frac{sin \frac{10 \pi }{11}}{32 \times sin ⁡ \frac{\pi }{11}}\right)^{2}=\frac{1}{\left(32\right)^{2}}=\frac{1}{1024}$