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  4. The value of cos (π/7) + cos (2π/7) + cos (3π/7) +cos (4π/7) + cos (5π/7) +cos (6π/7) + cos (7π/7) is

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Q. The value of $cos \frac{\pi}{7} + cos \frac{2\pi}{7} + cos \frac{3\pi}{7} +cos \frac{4\pi}{7} + cos \frac{5\pi}{7} +cos \frac{6\pi}{7} + cos \frac{7\pi}{7} $ is

UPSEEUPSEE 2015

Solution:

We have,
$\cos \frac{\pi}{7}+\cos \frac{2 \pi}{7}+\cos \frac{3 \pi}{7}+\cos \frac{4 \pi}{7}+\cos \frac{5 \pi}{7}$
$+\cos \frac{6 \pi}{7}+\cos \frac{7 \pi}{7}$
$=\left(\cos \frac{\pi}{7}+\cos \frac{6 \pi}{7}\right)+\left(\cos \frac{2 \pi}{7}+\cos \frac{5 \pi}{7}\right)$
$+\left(\cos \frac{3 \pi}{7}+\cos \frac{4 \pi}{7}\right)+\cos \pi$
$=\left(\cos \frac{\pi}{7}+\cos \left(\pi-\frac{\pi}{7}\right)\right)+\left(\cos \frac{2 \pi}{7}+\cos \left(\pi-\frac{2 \pi}{7}\right)\right)$
$+\left(\cos \frac{3 \pi}{7}+\cos \left(\pi-\frac{3 \pi}{7}\right)\right)+(-1)$
$=0+0+0-1=-1$