{sin ^4}frac{pi }{8} + {sin ^4}frac{{3pi }}{8 | vedclass

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(C) Given expression: $S = {\sin ^4}\frac{\pi }{8} + {\sin ^4}\frac{{3\pi }}{8} + {\sin ^4}\frac{{5\pi }}{8} + {\sin ^4}\frac{{7\pi }}{8}$Since $\sin(

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$\frac{3}{2}$

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(C) Given expression: $S = {\sin ^4}\frac{\pi }{8} + {\sin ^4}\frac{{3\pi }}{8} + {\sin ^4}\frac{{5\pi }}{8} + {\sin ^4}\frac{{7\pi }}{8}$Since $\sin(\pi - 	heta) = \sin 	heta$,we have $\sin \frac{{7\pi }}{8} = \sin \frac{\pi }{8}$ and $\sin \frac{{5\pi }}{8} = \sin \frac{{3\pi }}{8}$.So,$S = 2\left( {\sin ^4}\frac{\pi }{8} + {\sin ^4}\frac{{3\pi }}{8} ight)$.Using $2\sin^2 	heta = 1 - \cos 2	heta$,we have $S = 2 \left[ \frac{1}{4} (1 - \cos \frac{\pi }{4})^2 + \frac{1}{4} (1 - \cos \frac{{3\pi }}{4})^2 ight]$.$S = \frac{1}{2} \left[ (1 - \frac{1}{\sqrt{2}})^2 + (1 - (- \frac{1}{\sqrt{2}}))^2 ight]$.$S = \frac{1}{2} \left[ (1 - \frac{1}{\sqrt{2}})^2 + (1 + \frac{1}{\sqrt{2}})^2 ight]$.Using $(a-b)^2 + (a+b)^2 = 2(a^2 + b^2)$,we get $S = \frac{1}{2} \cdot 2 \left[ 1^2 + (\frac{1}{\sqrt{2}})^2 ight]$.$S = 1 + \frac{1}{2} = \frac{3}{2}$.

${\sin ^4}\frac{\pi }{8} + {\sin ^4}\frac{{3\pi }}{8} + {\sin ^4}\frac{{5\pi }}{8} + {\sin ^4}\frac{{7\pi }}{8} = $

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