The value of cos frac{pi}{7} cos frac{2pi}{7} | vedclass

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Question

(A) Let $P = \cos \frac{\pi}{7} \cos \frac{2\pi}{7} \cos \frac{3\pi}{7}$.Since $\cos \frac{3\pi}{7} = \cos (\pi - \frac{4\pi}{7}) = -\cos \frac{4\pi}{

Vedclass · Accepted Answer

$1/8$

Vedclass · Answer

(A) Let $P = \cos \frac{\pi}{7} \cos \frac{2\pi}{7} \cos \frac{3\pi}{7}$.Since $\cos \frac{3\pi}{7} = \cos (\pi - \frac{4\pi}{7}) = -\cos \frac{4\pi}{7}$,we have:$P = -\cos \frac{\pi}{7} \cos \frac{2\pi}{7} \cos \frac{4\pi}{7}$.Using the formula $\cos 	heta \cos 2	heta \cos 4	heta = \frac{\sin 8	heta}{8 \sin 	heta}$ for $	heta = \frac{\pi}{7}$:$P = -\left[ \frac{\sin (8 \cdot \frac{\pi}{7})}{8 \sin \frac{\pi}{7}} ight]$.Since $\sin \frac{8\pi}{7} = \sin (\pi + \frac{\pi}{7}) = -\sin \frac{\pi}{7}$:$P = -\left[ \frac{-\sin \frac{\pi}{7}}{8 \sin \frac{\pi}{7}} ight] = -\left( -\frac{1}{8} ight) = \frac{1}{8}$.

The value of $\cos \frac{\pi}{7} \cos \frac{2\pi}{7} \cos \frac{3\pi}{7}$ is

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