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  4. (1+ cos (π/8))(1+ cos (3 π/8))(1+ cos (5 π/8))(1+ cos (7 π/8)) is equal to

Q. $\left(1+\cos \frac{\pi}{8}\right)\left(1+\cos \frac{3 \pi}{8}\right)\left(1+\cos \frac{5 \pi}{8}\right)\left(1+\cos \frac{7 \pi}{8}\right)$ is equal to

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Solution:

$\left(1+\cos \frac{\pi}{8}\right)\left(1-\cos \frac{\pi}{8}\right)\left(1+\cos \frac{3 \pi}{8}\right)\left(1-\cos \frac{3 \pi}{8}\right)=\sin ^{2}\left(\frac{\pi}{8}\right) \sin ^{2}\left(\frac{3 \pi}{8}\right)$
$=\frac{\left(1-\cos \frac{\pi}{4}\right)}{2} \cdot \frac{\left(1-\cos \frac{3 \pi}{4}\right)}{2}=\frac{1}{8}$

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