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

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Question

(D) Given expression: $E = \sin ^4 \frac{\pi}{8} + \cos ^4 \frac{3 \pi}{8} - \sin ^4 \frac{3 \pi}{8} + \sin ^4 \frac{5 \pi}{8} + \cos ^4 \frac{7 \pi}{

Vedclass · Accepted Answer

$\frac{3}{4}$

Vedclass · Answer

(D) Given expression: $E = \sin ^4 \frac{\pi}{8} + \cos ^4 \frac{3 \pi}{8} - \sin ^4 \frac{3 \pi}{8} + \sin ^4 \frac{5 \pi}{8} + \cos ^4 \frac{7 \pi}{8} - \sin ^4 \frac{7 \pi}{8}$ Using $\sin(\pi - 	heta) = \sin 	heta$ and $\cos(\pi - 	heta) = -\cos 	heta$: $\sin^4 \frac{5 \pi}{8} = \sin^4(\pi - \frac{3 \pi}{8}) = \sin^4 \frac{3 \pi}{8}$ $\cos^4 \frac{7 \pi}{8} = \cos^4(\pi - \frac{\pi}{8}) = (-\cos \frac{\pi}{8})^4 = \cos^4 \frac{\pi}{8}$ $\sin^4 \frac{7 \pi}{8} = \sin^4(\pi - \frac{\pi}{8}) = \sin^4 \frac{\pi}{8}$ Substituting these into the expression: $E = \sin ^4 \frac{\pi}{8} + \cos ^4 \frac{3 \pi}{8} - \sin ^4 \frac{3 \pi}{8} + \sin ^4 \frac{3 \pi}{8} + \cos ^4 \frac{\pi}{8} - \sin ^4 \frac{\pi}{8}$ $E = \cos ^4 \frac{3 \pi}{8} + \cos ^4 \frac{\pi}{8}$ Using $\cos^2 	heta = \frac{1 + \cos 2 	heta}{2}$: $E = \left(\frac{1 + \cos \frac{3 \pi}{4}}{2}ight)^2 + \left(\frac{1 + \cos \frac{\pi}{4}}{2}ight)^2$ $E = \frac{1}{4} \left[ (1 - \frac{1}{\sqrt{2}})^2 + (1 + \frac{1}{\sqrt{2}})^2 ight]$ $E = \frac{1}{4} \left[ (1 + \frac{1}{2} - \sqrt{2}) + (1 + \frac{1}{2} + \sqrt{2}) ight]$ $E = \frac{1}{4} [3] = \frac{3}{4}$

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

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