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Introduction

Prove that 2c...

Question

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

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Solution

$L . H . S . = 2 cos \frac{π}{13} cos \frac{9 π}{13} + cos \frac{3 π}{13} + cos \frac{5 π}{13} = cos (\frac{9 π}{13} + \frac{π}{13}) + cos (\frac{9 π}{13} - \frac{π}{13}) + cos \frac{3 π}{13} + cos \frac{5 π}{13} = cos (\frac{10 π}{13}) + cos (\frac{8 π}{13}) + cos \frac{3 π}{13} + cos \frac{5 π}{13} = cos (\frac{10 π}{13}) + cos (\frac{8 π}{13}) + cos (π - \frac{10 π}{13}) + cos (π - \frac{8 π}{13})$
$= cos (\frac{10 π}{13}) + cos (\frac{8 π}{13}) - cos (\frac{10 π}{13}) - cos (\frac{8 π}{13}) {∵ cos (π - x) = - cos x}$
$= 0 = R . H . S .$

Hence proved.

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