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3.Trigonometrical Ratios, Functions and Identities
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The value of $2 \sin(\frac{\pi}{8}) \sin (\frac{2 \pi}{8}) \sin (\frac{3 \pi}{8}) \sin (\frac{5 \pi}{8}) \sin (\frac{6 \pi}{8}) \sin (\frac{7 \pi}{8})$ is:
A
$\frac{1}{4 \sqrt{2}}$
B
$\frac{1}{4}$
C
$\frac{1}{8}$
D
$\frac{1}{8 \sqrt{2}}$
(JEE MAIN-2021)
Solution
$2 \sin \left(\frac{\pi}{8}\right) \sin \left(\frac{2 \pi}{8}\right) \sin \left(\frac{3 \pi}{8}\right) \sin \left(\frac{5 \pi}{8}\right) \sin \left(\frac{6 \pi}{8}\right) \sin \left(\frac{7 \pi}{8}\right)$
$2 \sin ^{2} \frac{\pi}{8} \sin ^{2} \frac{2 \pi}{8} \sin ^{2} \frac{3 \pi}{8}$
$\sin ^{2} \frac{\pi}{8} \sin ^{2} \frac{3 \pi}{8}$
$\sin ^{2} \frac{\pi}{8} \cos ^{2} \frac{\pi}{8}$
$\frac{1}{4} \sin ^{2}\left(\frac{\pi}{4}\right)=\frac{1}{8}$
Standard 11
Mathematics
