Question

Prove that:
$2cos \frac{\pi}{13} cos \frac{9\pi}{13} + cos \frac{3\pi}{13} + cos \frac{5\pi}{13} = 0$


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Answer 1

Step-by-step explanation:

We have,

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

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

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

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

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

$= 0$