sum_{k=0}^4 sin^2 left( (2k+1) frac{pi}{20} r | vedclass

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(B) Let $S = \sum_{k=0}^4 \sin^2 \left( (2k+1) \frac{\pi}{20} \right)$.Expanding the summation,we get:$S = \sin^2 \frac{\pi}{20} + \sin^2 \frac{3\pi}{

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$\frac{5}{2}$

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(B) Let $S = \sum_{k=0}^4 \sin^2 \left( (2k+1) \frac{\pi}{20} ight)$.Expanding the summation,we get:$S = \sin^2 \frac{\pi}{20} + \sin^2 \frac{3\pi}{20} + \sin^2 \frac{5\pi}{20} + \sin^2 \frac{7\pi}{20} + \sin^2 \frac{9\pi}{20}$.Using the identity $\sin^2 	heta = \cos^2 \left( \frac{\pi}{2} - 	heta ight)$:$\sin^2 \frac{\pi}{20} = \cos^2 \left( \frac{\pi}{2} - \frac{\pi}{20} ight) = \cos^2 \frac{9\pi}{20}$.$\sin^2 \frac{3\pi}{20} = \cos^2 \left( \frac{\pi}{2} - \frac{3\pi}{20} ight) = \cos^2 \frac{7\pi}{20}$.Substituting these into the sum:$S = \cos^2 \frac{9\pi}{20} + \cos^2 \frac{7\pi}{20} + \sin^2 \frac{5\pi}{20} + \sin^2 \frac{7\pi}{20} + \sin^2 \frac{9\pi}{20}$.Grouping terms:$S = (\sin^2 \frac{9\pi}{20} + \cos^2 \frac{9\pi}{20}) + (\sin^2 \frac{7\pi}{20} + \cos^2 \frac{7\pi}{20}) + \sin^2 \frac{\pi}{4}$.Since $\sin^2 	heta + \cos^2 	heta = 1$ and $\sin \frac{\pi}{4} = \frac{1}{\sqrt{2}}$:$S = 1 + 1 + \left( \frac{1}{\sqrt{2}} ight)^2 = 2 + \frac{1}{2} = \frac{5}{2}$.

$\sum_{k=0}^4 \sin^2 \left( (2k+1) \frac{\pi}{20} \right) =$

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