sin frac{pi}{16} sin frac{3 pi}{16} sin frac{ | vedclass

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Question

(A) We have,$\sin \frac{\pi}{16} \sin \frac{3 \pi}{16} \sin \frac{5 \pi}{16} \sin \frac{7 \pi}{16}$ $= \sin \frac{\pi}{16} \sin \frac{3 \pi}{16} \sin

Vedclass · Accepted Answer

$\frac{\sqrt{2}}{16}$

Vedclass · Answer

(A) We have,$\sin \frac{\pi}{16} \sin \frac{3 \pi}{16} \sin \frac{5 \pi}{16} \sin \frac{7 \pi}{16}$ $= \sin \frac{\pi}{16} \sin \frac{3 \pi}{16} \sin \left(\frac{\pi}{2} - \frac{3 \pi}{16}\right) \sin \left(\frac{\pi}{2} - \frac{\pi}{16}\right)$ $= \sin \frac{\pi}{16} \sin \frac{3 \pi}{16} \cos \frac{3 \pi}{16} \cos \frac{\pi}{16}$ $= \frac{1}{4} \left(2 \sin \frac{\pi}{16} \cos \frac{\pi}{16} \cdot 2 \sin \frac{3 \pi}{16} \cos \frac{3 \pi}{16}\right)$ $= \frac{1}{4} \left(\sin \frac{\pi}{8} \sin \frac{3 \pi}{8}\right) = \frac{1}{4} \left(\sin \frac{\pi}{8} \sin \left(\frac{\pi}{2} - \frac{\pi}{8}\right)\right)$ $= \frac{1}{4} \sin \frac{\pi}{8} \cos \frac{\pi}{8} = \frac{1}{8} \sin \frac{\pi}{4} = \frac{1}{8 \sqrt{2}} = \frac{\sqrt{2}}{16}$

List-$A$	List-$B$
$(I)$ $\tan \left(\frac{\alpha + \beta}{2}\right) =$	$(a)$ $\frac{b}{a}$
$(II)$ $\cos (\alpha + \beta) =$	$(b)$ $\frac{2ab}{a^2 + b^2}$
$(III)$ $\sin (\alpha + \beta) =$	$(c)$ $\frac{2ab}{a^2 - b^2}$
$(IV)$ $\tan (\alpha + \beta) =$	$(d)$ $\frac{a^2 - b^2}{a^2 + b^2}$

$\sin \frac{\pi}{16} \sin \frac{3 \pi}{16} \sin \frac{5 \pi}{16} \sin \frac{7 \pi}{16}$ is equal to

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