Question

༒ Bʀᴀɪɴʟʏ Sᴛᴀʀs

༒Mᴏᴅᴇʀᴀᴛᴇʀs

༒Other best user
$\pink{2cos \frac{\pi}{13} \cos \frac{9\pi}{13}+ \cos \frac{3\pi}{13} + \cos \frac{5\pi}{13} = 0}$
No spam
Solve it

Spammers Go Away​

Answer 1

__________________________________________

$\huge\fcolorbox{pink}{Purple}{ ★Question✍︎}$

2cos 13 π cos 13 9π +cos 13 3π +cos 13 5π =0

$\huge\fcolorbox{pink}{Purple}{ ★Answer ✍︎}$

L.H.S

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

Using that:-

$2 \cos \: a \: \cos \: b \: = \: (a + b) + \cos(a - b)$

$2 \cos \: \frac{\pi}{13} \cos \: \frac{9\pi}{13} + \cos \: \frac{3\pi}{13} + \cos \: \frac{5\pi}{13}$

$= \cos \: ( \frac{\pi}{13} + \frac{9\pi}{13} ) \: + \: \cos( \frac{\pi}{13} - \frac{9\pi}{13} ) + \cos \: \frac{3\pi}{13} + \cos \: \frac{5\pi}{13}$

$= \cos \: \frac{10\pi}{13} + \cos \: ( \frac{8\pi}{13} ) + \cos \: \frac{3\pi}{13} + \cos \: \frac{5\pi}{13} \cos \: (0 - ) \: = \: \cos \: 0$

$= \cos \: \frac{10\pi}{13 } + \: \cos \: ( \frac{ - 8\pi}{13} ) + \: \cos \: \frac{3\pi}{13} + \cos \: \frac{5\pi}{13}$

$= \: \cos \: (\pi - \frac{3\pi}{13} ) + \cos \: (\pi - \frac{5\pi}{13} ) + \cos \: \frac{3\pi}{13} + \cos \: \frac{5\pi}{13} \:$

$= - \cos \: \frac{3\pi}{13} - \cos \: \frac{5\pi}{13} + \cos \: \frac{3\pi}{13} + \cos \: \frac{5\pi}{13}$

$= \: 0.$

R.HS

__________________________________________

$\huge\fcolorbox{pink}{green}{ ★A᭄ΠSWΣRed by @MissFairy01 }$