Question

State true or false.

If $A+B+C=\pi$, then :$\cos \frac{2\pi }{7}+\cos \frac{4\pi }{7}+\cos \frac{6\pi }{7}=-\frac{1}{4}$.

• A. True
• B. False

Answer 1

The correct option is B False

$\cos \left(\frac{2\pi }{7}\right)+\cos \left(\frac{4\pi }{7}\right)+\cos \left(\frac{6\pi }{7}\right)$

Multiply and divide by $\sin \left(\frac{\pi }{7}\right)$

$=\frac{\sin \left(\frac{\pi }{7}\right)\cos \left(\frac{2\pi }{7}\right)+\sin \left(\frac{\pi }{7}\right)\cos \left(\frac{4\pi }{7}\right)+\sin \left(\frac{\pi }{7}\right)\cos \left(\frac{6\pi }{7}\right)}{\sin \left(\frac{\pi }{7}\right)}$

$=\frac{\sin (\frac{\pi }{7}-\frac{2\pi }{7})+\sin (\frac{\pi }{7}+\frac{2\pi }{7})+\sin (\frac{\pi }{7}-\frac{4\pi }{7})+\sin (\frac{\pi }{7}+\frac{4\pi }{7})+\sin (\frac{\pi }{7}-\frac{6\pi }{7})+\sin (\frac{\pi }{7}+\frac{6\pi }{7})}{2\sin \frac{\pi }{7}}$

$=\frac{\sin (-\frac{\pi }{7})+\sin \left(\frac{3\pi }{7}\right)+\sin (-\frac{3\pi }{7})+\sin \left(\frac{5\pi }{7}\right)+\sin (-\frac{5\pi }{7})+\sin \left(\frac{7\pi }{7}\right)}{2\sin \frac{\pi }{7}}$

$=\frac{-\sin \left(\frac{\pi }{7}\right)+\sin \left(\frac{3\pi }{7}\right)-\sin \left(\frac{3\pi }{7}\right)+\sin \left(\frac{5\pi }{7}\right)-\sin \left(\frac{5\pi }{7}\right)+\sin \left(\frac{7\pi }{7}\right)}{2\sin \frac{\pi }{7}}$

$=\frac{-\sin \left(\frac{\pi }{7}\right)+\sin \pi }{2\sin \frac{\pi }{7}}$

$=\frac{-1}{2}$

The given statement is false.