4 cos frac{pi}{7} cos frac{pi}{5} cos frac{2 | vedclass

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Question

(A) We know the identity $\cos 	heta \cos 2	heta \cos 4	heta = \frac{\sin 8	heta}{8 \sin 	heta}$.First,consider the product $P_1 = \cos \frac{\pi

Vedclass · Accepted Answer

$-\frac{1}{8}$

Vedclass · Answer

(A) We know the identity $\cos 	heta \cos 2	heta \cos 4	heta = \frac{\sin 8	heta}{8 \sin 	heta}$.First,consider the product $P_1 = \cos \frac{\pi}{7} \cos \frac{2 \pi}{7} \cos \frac{4 \pi}{7}$.Using the identity with $	heta = \frac{\pi}{7}$,we get $P_1 = \frac{\sin(8\pi/7)}{8 \sin(\pi/7)} = \frac{\sin(\pi + \pi/7)}{8 \sin(\pi/7)} = \frac{-\sin(\pi/7)}{8 \sin(\pi/7)} = -\frac{1}{8}$.Next,consider the product $P_2 = \cos \frac{\pi}{5} \cos \frac{2 \pi}{5}$.Using the identity $\cos 	heta \cos 2	heta = \frac{\sin 4	heta}{4 \sin 	heta}$,we get $P_2 = \frac{\sin(4\pi/5)}{4 \sin(\pi/5)} = \frac{\sin(\pi - \pi/5)}{4 \sin(\pi/5)} = \frac{\sin(\pi/5)}{4 \sin(\pi/5)} = \frac{1}{4}$.Therefore,the total expression is $4 	imes P_1 	imes P_2 = 4 	imes (-\frac{1}{8}) 	imes (\frac{1}{4}) = -\frac{1}{8}$.

$4 \cos \frac{\pi}{7} \cos \frac{\pi}{5} \cos \frac{2 \pi}{7} \cos \frac{2 \pi}{5} \cos \frac{4 \pi}{7} = $

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