cos frac{pi}{7} cos frac{2pi}{7} cos frac{3pi | vedclass

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Question

(C) Let $P = \cos \frac{\pi}{7} \cos \frac{2\pi}{7} \cos \frac{3\pi}{7}$.Multiply and divide by $2 \sin \frac{\pi}{7}$:$P = \frac{1}{2 \sin \frac{\pi}

Vedclass · Accepted Answer

$\frac{1}{8}$

Vedclass · Answer

(C) Let $P = \cos \frac{\pi}{7} \cos \frac{2\pi}{7} \cos \frac{3\pi}{7}$.Multiply and divide by $2 \sin \frac{\pi}{7}$:$P = \frac{1}{2 \sin \frac{\pi}{7}} \left( 2 \sin \frac{\pi}{7} \cos \frac{\pi}{7} ight) \cos \frac{2\pi}{7} \cos \frac{3\pi}{7}$Using $2 \sin 	heta \cos 	heta = \sin 2	heta$:$P = \frac{1}{2 \sin \frac{\pi}{7}} \left( \sin \frac{2\pi}{7} \cos \frac{2\pi}{7} ight) \cos \frac{3\pi}{7}$Multiply and divide by $2$:$P = \frac{1}{4 \sin \frac{\pi}{7}} \left( 2 \sin \frac{2\pi}{7} \cos \frac{2\pi}{7} ight) \cos \frac{3\pi}{7} = \frac{1}{4 \sin \frac{\pi}{7}} \sin \frac{4\pi}{7} \cos \frac{3\pi}{7}$Since $\sin \frac{4\pi}{7} = \sin \left( \pi - \frac{3\pi}{7} ight) = \sin \frac{3\pi}{7}$:$P = \frac{1}{4 \sin \frac{\pi}{7}} \sin \frac{3\pi}{7} \cos \frac{3\pi}{7} = \frac{1}{8 \sin \frac{\pi}{7}} \left( 2 \sin \frac{3\pi}{7} \cos \frac{3\pi}{7} ight)$$P = \frac{\sin \frac{6\pi}{7}}{8 \sin \frac{\pi}{7}}$Since $\sin \frac{6\pi}{7} = \sin \left( \pi - \frac{\pi}{7} ight) = \sin \frac{\pi}{7}$:$P = \frac{\sin \frac{\pi}{7}}{8 \sin \frac{\pi}{7}} = \frac{1}{8}$.

$\cos \frac{\pi}{7} \cos \frac{2\pi}{7} \cos \frac{3\pi}{7} =$

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