# Write the value of cos

Question:

Write the value of $\cos \frac{\pi}{7} \cos \frac{2 \pi}{7} \cos \frac{4 \pi}{7}$.

Solution:

We have,

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

$\left[\right.$ On dividing and multiplying by $\left.2 \sin \frac{\pi}{7}\right]$

$=\frac{2 \times \sin \frac{2 \pi}{7} \cos \frac{2 \pi}{7} \cos \frac{4 \pi}{7}}{2 \times 2 \sin \frac{\pi}{7}}$

Proceeding in the same way, we get

$\cos \frac{\pi}{7} \cos \frac{2 \pi}{7} \cos \frac{4 \pi}{7}=\frac{\sin \frac{8 \pi}{7}}{8 \sin \frac{\pi}{7}}$

$=\frac{\sin \left(\pi+\frac{\pi}{7}\right)}{8 \sin \frac{\pi}{7}}$

$=\frac{-\sin \frac{\pi}{7}}{8 \sin \frac{\pi}{7}}$

$\therefore \cos \frac{\pi}{7} \cos \frac{2 \pi}{7} \cos \frac{4 \pi}{7}=\frac{-1}{8}$

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