sin ^2 frac{pi }{8} + sin ^2 frac{3pi }{8} + | vedclass

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(D) Given expression: $S = \sin ^2 \frac{\pi }{8} + \sin ^2 \frac{3\pi }{8} + \sin ^2 \frac{5\pi }{8} + \sin ^2 \frac{7\pi }{8}$Using the identity $\s

Vedclass · Accepted Answer

$2$

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(D) Given expression: $S = \sin ^2 \frac{\pi }{8} + \sin ^2 \frac{3\pi }{8} + \sin ^2 \frac{5\pi }{8} + \sin ^2 \frac{7\pi }{8}$Using the identity $\sin(\pi - 	heta) = \sin 	heta$,we have:$\sin \frac{5\pi }{8} = \sin(\pi - \frac{3\pi }{8}) = \sin \frac{3\pi }{8}$$\sin \frac{7\pi }{8} = \sin(\pi - \frac{\pi }{8}) = \sin \frac{\pi }{8}$Substituting these into the expression:$S = \sin ^2 \frac{\pi }{8} + \sin ^2 \frac{3\pi }{8} + \sin ^2 \frac{3\pi }{8} + \sin ^2 \frac{\pi }{8}$$S = 2(\sin ^2 \frac{\pi }{8} + \sin ^2 \frac{3\pi }{8})$Using $\sin^2 	heta + \sin^2(\frac{\pi}{2} - 	heta) = \sin^2 	heta + \cos^2 	heta = 1$:Since $\frac{3\pi}{8} = \frac{\pi}{2} - \frac{\pi}{8}$,then $\sin \frac{3\pi}{8} = \cos \frac{\pi}{8}$.$S = 2(\sin ^2 \frac{\pi }{8} + \cos ^2 \frac{\pi }{8}) = 2(1) = 2$.

$\sin ^2 \frac{\pi }{8} + \sin ^2 \frac{3\pi }{8} + \sin ^2 \frac{5\pi }{8} + \sin ^2 \frac{7\pi }{8} = $

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