Question

cos{π/13).cos(9π/13)+cos{3π/13)+cos(5π/13)=0, proof it

Answer 1

First of all the question is incorrect as the first cos part include a 2
The ans is as below

Answer 2

$\bold{\huge{\underline{Question}}}$

2cos{π/13).cos(9π/13)+cos{3π/13)+cos(5π/13)=0, proof it

$\bold{\huge{\underline{Solution-}}}$

LHS = 2 cos{π/13).cos(9π/13)+cos{3π/13)+cos(5π/13)

$\bf \: [2 \: cos \frac{9\pi}{13} \times cos \frac{\pi}{13} ] + cos \frac{3\pi}{13} + cos \frac{5\pi}{13} \\ \\ \bf \: [ cos\{ \frac{9\pi}{13} + \frac{\pi}{13} \} + cos \{ \frac{9\pi}{13 - \frac{\pi}{13} } \}] + cos \frac{3\pi}{13} + cos \frac{5\pi}{13} \\ \\ \bf \: cos \frac{10\pi}{13} + cos \frac{8\pi}{13} + cos \frac{3\pi}{13} + cos \frac{5\pi}{13} \\ \\ \bf \: cos[\pi - \frac{3\pi}{13} ] + cos[\pi - \frac{5\pi}{13} ] + cos \frac{3\pi}{13} + cos \frac{5\pi}{13} \\ \\ \bf \: - cos \frac{3\pi}{13} + ( - cos \frac{5\pi}{13} ) + cos \frac{3\pi}{13} + cos \frac{5\pi}{13} \\ \\ \bf \: = 0$