Question

$2cos \frac{π}{13}cos\frac{9π}{13}+cos \frac{3π}{13}+cos \frac{5π}{13}=0$

Answer 1

L.H.S=$$ $2cos \frac{π}{13}cos\frac{9π}{13}+cos \frac{3π}{13}+cos \frac{5π}{13}$

$=2cos \frac{π}{13}cos\frac{9π}{13}+2cos(\frac{\frac{3{\pi}}{13}+\frac{5{\pi}}{13}}{2})cos(\frac{3{\pi}{13}-\frac{5\pi}{13}}{2})$ $[cos\,x+cos\,y=2cos(\frac{x+y}{2})cos(\frac{x-y}{2})]$

$=2cos \frac{π}{13}cos\frac{9π}{13}+2cos\frac{4{\pi}}{13}cos(\frac{-\pi}{13})$

$=2cos \frac{π}{13}cos\frac{9π}{13}+2cos\frac{4{\pi}}{13}cos\frac{\pi}{13}$

$=2cos \frac{π}{13}[cos\frac{9π}{13}+2cos\frac{4{\pi}}{13}]$

$=2cos \frac{π}{13}[2cos(\frac{\frac{9π}{13}+\frac{4{\pi}}{13}}{2})cos(\frac{\frac{9{\pi}}{13}+\frac{4{\pi}}{13}}{2})]$

$2cos\frac{\pi}{13}[2cos\frac{\pi}{2}cos\frac{5{\pi}}{26}]$

$2cos\frac{\pi}{13}×2×0×cos\frac{5{\pi}}{26}$

$= 0 = R.H.S$