First slide

Trigonometric transformations

Question

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

Moderate

A

-1
B

0
C

1
D

$\frac{\sqrt{3}}{2}$

Solution

$\begin{array}{l} 2 \cos \frac{π}{13} \cos \frac{9 π}{13} + \cos \frac{3 π}{13} + \cos \frac{5 π}{13} \\ = 2 \cos \frac{π}{13} \cos \frac{9 π}{13} + 2 \cos \frac{π}{13} \cos \frac{4 π}{13} \\ = 2 \cos \frac{π}{13} [\cos (\frac{9 π}{13}) + \cos (\frac{4 π}{13})] \\ = 2 \cos (\frac{π}{13}) [2 \cos \frac{π}{2} \cos \frac{5 π}{26}] = 0 \end{array}$

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