cos left(frac{pi}{7}right) cos left(frac{2 pi | vedclass

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Question

(A) Let $P = \cos \left(\frac{\pi}{7}\right) \cos \left(\frac{2 \pi}{7}\right) \cos \left(\frac{4 \pi}{7}\right)$.Multiply and divide by $2 \sin \left

Vedclass · Accepted Answer

$\frac{-1}{8}$

Vedclass · Answer

(A) Let $P = \cos \left(\frac{\pi}{7}ight) \cos \left(\frac{2 \pi}{7}ight) \cos \left(\frac{4 \pi}{7}ight)$.Multiply and divide by $2 \sin \left(\frac{\pi}{7}ight)$:$P = \frac{2 \sin \left(\frac{\pi}{7}ight) \cos \left(\frac{\pi}{7}ight) \cos \left(\frac{2 \pi}{7}ight) \cos \left(\frac{4 \pi}{7}ight)}{2 \sin \left(\frac{\pi}{7}ight)}$Using $2 \sin 	heta \cos 	heta = \sin 2 	heta$:$P = \frac{\sin \left(\frac{2 \pi}{7}ight) \cos \left(\frac{2 \pi}{7}ight) \cos \left(\frac{4 \pi}{7}ight)}{2 \sin \left(\frac{\pi}{7}ight)}$Multiply numerator and denominator by $2$:$P = \frac{2 \sin \left(\frac{2 \pi}{7}ight) \cos \left(\frac{2 \pi}{7}ight) \cos \left(\frac{4 \pi}{7}ight)}{4 \sin \left(\frac{\pi}{7}ight)} = \frac{\sin \left(\frac{4 \pi}{7}ight) \cos \left(\frac{4 \pi}{7}ight)}{4 \sin \left(\frac{\pi}{7}ight)}$Multiply numerator and denominator by $2$ again:$P = \frac{2 \sin \left(\frac{4 \pi}{7}ight) \cos \left(\frac{4 \pi}{7}ight)}{8 \sin \left(\frac{\pi}{7}ight)} = \frac{\sin \left(\frac{8 \pi}{7}ight)}{8 \sin \left(\frac{\pi}{7}ight)}$Since $\sin \left(\frac{8 \pi}{7}ight) = \sin \left(\pi + \frac{\pi}{7}ight) = -\sin \left(\frac{\pi}{7}ight)$:$P = \frac{-\sin \left(\frac{\pi}{7}ight)}{8 \sin \left(\frac{\pi}{7}ight)} = -\frac{1}{8}$.

$\cos \left(\frac{\pi}{7}\right) \cos \left(\frac{2 \pi}{7}\right) \cos \left(\frac{4 \pi}{7}\right) = $

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