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  4. The value of cos (π/5) cos (2 π/5) cos (4 π/5) cos (8 π/5) is equal to

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Q. The value of $\cos \frac{\pi}{5} \cos \frac{2 \pi}{5} \cos \frac{4 \pi}{5} \cos \frac{8 \pi}{5}$ is equal to

Trigonometric Functions

Solution:

We have,$\cos \frac{\pi}{5} \cos \frac{2 \pi}{5} \cos \frac{4 \pi}{5} \cos \frac{8 \pi}{5} $
$=\frac{1}{2 \sin \frac{\pi}{5}} 2 \sin \frac{\pi}{5} \cos \frac{\pi}{5} \cos \frac{2 \pi}{5} \cos \frac{4 \pi}{5} \cos \frac{8 \pi}{5} $
$=\frac{1}{2 \sin \frac{\pi}{5}} \sin \frac{2 \pi}{5} \cos \frac{2 \pi}{5} \cos \frac{4 \pi}{5} \cos \frac{8 \pi}{5}$
$=\frac{1}{4 \sin \frac{\pi}{5}} \sin \frac{4 \pi}{5} \cos \frac{4 \pi}{5} \cos \frac{8 \pi}{5}$
$=\frac{1}{8 \sin \frac{\pi}{5}} \sin \frac{8 \pi}{5} \cos \frac{8 \pi}{5}$
$=\frac{\sin \frac{16 \pi}{5}}{16 \sin \frac{\pi}{5}}$
$=\frac{\sin \left(3 \pi+\frac{\pi}{5}\right)}{16 \sin \frac{\pi}{5}}$
$=\frac{-\sin \frac{\pi}{5}}{16 \sin \frac{\pi}{5}}$
$=-\frac{1}{16}$