Question

Find the value of $\cos \frac{\pi }{11}+\cos \frac{3\pi }{11}+\cos \frac{5\pi }{11}+\cos \frac{7\pi }{11}+\cos \frac{9\pi }{11}$.

Answer 1

$S=\cos \frac{\pi }{11}+\cos \frac{3\pi }{11}+\cos \frac{5\pi }{11}+\cos \frac{7\pi }{11}+\cos \frac{9\pi }{11}$

Multiply both sides by $2\sin \pi /11$

$2\sin \frac{\pi }{11}s=2\sin \frac{\pi }{11}\cos \frac{\pi }{11}+2\sin \frac{\pi }{11}\cos \frac{3\pi }{11}+2\sin \frac{\pi }{11}\cos \frac{5\pi }{11}+2\sin \frac{\pi }{11}\cos \frac{7\pi }{11}+2\sin \frac{\pi }{11}\cos \frac{9\pi }{11}$

[we know $2\sin A\cos B=\sin (A+B)+\sin (A-B)]$

$\therefore 2\sin \frac{\pi }{11}s=\sin \frac{2\pi }{11}+\sin \frac{4\pi }{11}-\sin \frac{2\pi }{11}+\sin \frac{6\pi }{11}-\sin \frac{4\pi }{11}+\sin \frac{8\pi }{11}-\sin \frac{6\pi }{11}+\sin \frac{10\pi }{11}-\sin 8\pi 11$

$\therefore 2\sin \frac{\pi }{11}s=\sin \frac{10\pi }{11}$

$\Rightarrow S=\frac{\sin \frac{10\pi }{11}}{2\sin \frac{\pi }{11}}=\frac{\sin (\pi -\frac{\pi }{11})}{2\sin \pi /11}$

$=\frac{1}{2}\frac{\sin \pi /11}{\sin \pi /11}=\frac{1}{2}$.