cos ^4 frac{pi}{8}+cos ^4 frac{2 pi}{8}+cos ^ | vedclass

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(A) ધારો કે $S = \sum_{k=1}^{8} \cos^4 \frac{k\pi}{8}$.ગુણધર્મ $\cos(\pi - 	heta) = -\cos 	heta$ નો ઉપયોગ કરતા,$\cos^4(\pi - 	heta) = \cos^4 	heta

Vedclass · Accepted Answer

$3$

Vedclass · Answer

(A) ધારો કે $S = \sum_{k=1}^{8} \cos^4 \frac{k\pi}{8}$.ગુણધર્મ $\cos(\pi - 	heta) = -\cos 	heta$ નો ઉપયોગ કરતા,$\cos^4(\pi - 	heta) = \cos^4 	heta$ મળે.તેથી,$\cos^4 \frac{7\pi}{8} = \cos^4 \frac{\pi}{8}$,$\cos^4 \frac{6\pi}{8} = \cos^4 \frac{2\pi}{8}$,અને $\cos^4 \frac{5\pi}{8} = \cos^4 \frac{3\pi}{8}$.વળી,$\cos \frac{4\pi}{8} = 0$ અને $\cos \frac{8\pi}{8} = -1$.તેથી,$S = 2(\cos^4 \frac{\pi}{8} + \cos^4 \frac{2\pi}{8} + \cos^4 \frac{3\pi}{8}) + 0^4 + (-1)^4$.$S = 2(\cos^4 \frac{\pi}{8} + \sin^4 \frac{\pi}{8} + (\frac{1}{\sqrt{2}})^4) + 1$.$\cos^4 	heta + \sin^4 	heta = 1 - \frac{1}{2}\sin^2(2	heta)$ સૂત્ર વાપરતા:$S = 2(1 - \frac{1}{2} \cdot \frac{1}{2} + \frac{1}{4}) + 1 = 2(1) + 1 = 3$.

$\cos ^4 \frac{\pi}{8}+\cos ^4 \frac{2 \pi}{8}+\cos ^4 \frac{3 \pi}{8}+\cos ^4 \frac{4 \pi}{8}+\cos ^4 \frac{5 \pi}{8}+\cos ^4 \frac{6 \pi}{8}+\cos ^4 \frac{7 \pi}{8}+\cos ^4 \frac{8 \pi}{8}$ ની કિંમત શોધો.

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