The value of cos frac{2 pi}{15} cos frac{4 pi | vedclass

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Question

(A) Let $P = \cos \frac{2 \pi}{15} \cos \frac{4 \pi}{15} \cos \frac{8 \pi}{15} \cos \frac{14 \pi}{15}$.Note that $\frac{14 \pi}{15} = \pi - \frac{\pi}

Vedclass · Accepted Answer

$\frac{1}{16}$

Vedclass · Answer

(A) Let $P = \cos \frac{2 \pi}{15} \cos \frac{4 \pi}{15} \cos \frac{8 \pi}{15} \cos \frac{14 \pi}{15}$.Note that $\frac{14 \pi}{15} = \pi - \frac{\pi}{15}$,so $\cos \frac{14 \pi}{15} = -\cos \frac{\pi}{15}$.Thus,$P = -\cos \frac{\pi}{15} \cos \frac{2 \pi}{15} \cos \frac{4 \pi}{15} \cos \frac{8 \pi}{15}$.Using the formula $\cos 	heta \cos 2 	heta \cos 4 	heta \dots \cos 2^{n-1} 	heta = \frac{\sin 2^n 	heta}{2^n \sin 	heta}$ with $	heta = \frac{\pi}{15}$ and $n = 4$:$P = -\left( \frac{\sin(2^4 \cdot \frac{\pi}{15})}{2^4 \sin \frac{\pi}{15}} ight) = -\frac{\sin \frac{16 \pi}{15}}{16 \sin \frac{\pi}{15}}$.Since $\sin \frac{16 \pi}{15} = \sin(\pi + \frac{\pi}{15}) = -\sin \frac{\pi}{15}$,$P = -\left( \frac{-\sin \frac{\pi}{15}}{16 \sin \frac{\pi}{15}} ight) = \frac{1}{16}$.

The value of $\cos \frac{2 \pi}{15} \cos \frac{4 \pi}{15} \cos \frac{8 \pi}{15} \cos \frac{14 \pi}{15}$ is :

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