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Question Number 97658 by Vishal Sharma last updated on 09/Jun/20

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

Commented by bemath last updated on 09/Jun/20

x = \frac{2 π}{7}

\frac{\cos x + \cos 2 x + \cos 3 x}{2 \sin x} \times 2 \sin x

\frac{\sin 2 x + 2 \sin x \cos 2 x + 2 \sin x \cos 3 x}{2 \sin x} =

\frac{\sin 2 x + \sin 3 x - \sin x + \sin 4 x - \sin 2 x}{2 \sin x} =

\frac{\sin 4 x + \sin 3 x - \sin x}{2 \sin x} =

\frac{2 \sin (\frac{7 x}{2}) \cos (\frac{x}{2}) - \sin x}{2 \sin x} = - \frac{1}{2}

[\sin (\frac{7 x}{2}) = \sin π = 0]

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