Question

2cospi/13 cos9pi/13+cos3pi/13+cos5pi/13=0

Answer 1

Step-by-step explanation:

We have,

$2 \cos \bigg( \frac{\pi}{13} \bigg) \cos \bigg( \frac{9\pi}{13} \bigg) + \cos \bigg( \frac{3\pi}{13} \bigg) + \cos \bigg( \frac{5\pi}{13} \bigg)$

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

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

$= 2 \cos \bigg( \frac{\pi}{13} \bigg) . 2\cos \bigg( \frac{13\pi}{26} \bigg) \cos \bigg( \frac{5\pi}{26} \bigg)$

$= 4 \cos \bigg( \frac{\pi}{13} \bigg) \cos \bigg( \frac{\pi}{2} \bigg) \cos \bigg( \frac{5\pi}{26} \bigg)$

$= 4 \cos \bigg( \frac{\pi}{13} \bigg).(0). \cos \bigg( \frac{5\pi}{26} \bigg)$

$= 0$