The value of $\left(1+\cos \frac{\pi}{8}\right)\left(1+\cos \frac{3 \pi}{8}\right)\left(1+\cos \frac{5 \pi}{8}\right)\left(1+\cos \frac{7 \pi}{8}\right)$ is

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Aicharya · Accepted Answer

$\left(1+\cos \frac{\pi}{8}ight)\left(1+\cos \frac{7 \pi}{8}ight)\left(1+\cos \frac{3 \pi}{8}ight)\left(1+\cos \frac{5 \pi}{8}ight)$ $=\left(1+\cos \frac{\pi}{8}ight)\left(1-\cos \frac{\pi}{8}ight)\left(1+\cos \frac{3 \pi}{8}ight)\left(1-\cos \frac{3 \pi}{8}ight)$ $\ldots[\because \cos (\pi-	heta)=-\cos 	heta]$ $\begin{aligned} & =\left(1-\cos ^2 \frac{\pi}{8}ight)\left(1-\cos ^2 \frac{3 \pi}{8}ight) \ & =\sin ^2 \frac{\pi}{8} \sin ^2 \frac{3 \pi}{8} \ & =\frac{1}{4}\left(2 \sin \frac{\pi}{8} \cdot \sin \frac{3 \pi}{8}ight)^2 \ & =\frac{1}{4}\left(\cos \frac{\pi}{4}-\cos \frac{\pi}{2}ight)^2 \ & =\frac{1}{8}\end{aligned}$

The value of $\left(1+\cos \frac{\pi}{8}\right)\left(1+\cos \frac{3 \pi}{8}\right)\left(1+\cos \frac{5…

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