First slide

Multiple and sub- multiple Angles

Question

$The value of$ $\cos \frac{2 π}{7} + \cos \frac{4 π}{7} + \cos \frac{6 π}{7} is$

Moderate

A

0
B

1
C

$\frac{1}{2}$
D

$- \frac{1}{2}$

Solution

$\begin{array}{l} We have, \cos \frac{2 π}{7} + \cos \frac{4 π}{7} + \cos \frac{6 π}{7} \\ = \frac{1}{2 \sin \frac{π}{7}} [2 \sin \frac{π}{7} \cos \frac{2 π}{7} + 2 \sin \frac{π}{7} \cos \frac{4 π}{7} + 2 \sin \frac{π}{7} \cos \frac{6 π}{7}] \\ = \frac{1}{2 \sin \frac{π}{7}} [(\sin \frac{3 π}{7} - \sin \frac{π}{7}) + (\sin \frac{5 π}{7} - \sin \frac{3 π}{7}) + (\sin \frac{7 π}{7} - \sin \frac{5 π}{7})] \\ = - \frac{1}{2} \end{array}$

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