Question

heya!! good afternoon all....

Please solve this trigonometry question from class11 .

No spams allowed ❎

Question .
$prove \: that \: \\ 2 \cos \frac{\pi}{13} \cos \frac{9\pi}{13} + \cos \frac{3\pi}{13} + \cos \frac{5\pi}{13} = 0$
Please answer quick....

Best answer = brainliest

Answer 1

The answer is explained in the attachment.


Hope it helps!

Answer 2

$\large{\mathit{HELLO\:\: DEAR,}}$

$\mathit{PROVE \:\: THAT} \: \\ \mathit{2 \cos \frac{\pi}{13} \cos \frac{9\pi}{13} + \cos \frac{3\pi}{13} + \cos \frac{5\pi}{13} = 0}$

$\mathit{2cos\frac{\pi}{13}cos\frac{9\pi}{13} + cos\frac{3\pi}{13} + cos\frac{5\pi}{13}} \\ \\ \mathit{2cos\frac{\pi}{13}cos\frac{9\pi}{13} + [ cos\frac{5\pi}{13} + cos\frac{3\pi}{13}]}$

$\mathit{2cos\frac{\pi}{13}cos\frac{9\pi}{13} + [2cos\frac{\frac{5\pi}{13} + \frac{3\pi}{13}}{2}cos\frac{\frac{5\pi}{13} - \frac{3\pi}{13}}{2}]}$
$\mathit{\to\to\to\to\to\to\to\to\therefore\boxed{cosx + cosy = 2cos(\frac{x + y}{2})cos(\frac{x - y}{2})}}$

$\mathit{2cos\frac{\pi}{13}cos\frac{9\pi}{13} + 2cos\frac{\frac{8\pi}{13}}{2}cos\frac{\frac{2\pi}{13}}{2}}$

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

$\mathit{2cos\frac{\pi}{13}[cos\frac{9\pi}{13} + cos\frac{4\pi}{13}]}$
$\mathit{\to\to\to\to\to\to\to\to\therefore\boxed{cosx + cosy = 2cos(\frac{x + y}{2})cos(\frac{x - y}{2})}}$

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

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

$\mathit{2cos\frac{\pi}{13}[2*0*cos\frac{5\pi}{26}]} \\ \mathit{\to\to\to\to\to\to\to\to\therefore\boxed{cos\frac{\pi}{2} = 0}}$

$\mathit{=0} \\ \\ \mathit{HENCE,} \\ \\ \mathit{2 \cos \frac{\pi}{13} \cos \frac{9\pi}{13} + \cos \frac{3\pi}{13} + \cos \frac{5\pi}{13} = 0}$

$\large{\mathit{\underline{I \: \: HOPE \: \: ITS \: \: HELP \: \: YOU \: \: DEAR, \: \: THANKS}}}$