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Question Number 128285 by john_santu last updated on 06/Jan/21

\cos (\frac{π}{7}) - \cos (\frac{2 π}{7}) + \cos (\frac{3 π}{7}) = ?

Answered by liberty last updated on 06/Jan/21

let φ = \cos (\frac{π}{7}) - \cos (\frac{2 π}{7}) + \cos (\frac{3 π}{7})

φ = \cos (\frac{π}{7}) + \cos (\frac{3 π}{7}) + \cos (\frac{5 π}{7})

\Rightarrow 2 φ \sin t = 2 \sin t \cos t + 2 \sin t \cos 3 t + 2 \sin tcos 5 t

where t = \frac{π}{7}

\Rightarrow 2 φ \sin t = \sin 2 t + \sin 4 t - \sin 2 t + \sin 6 t - \sin 4 t

\Rightarrow 2 φ \sin t = \sin 6 t

\Rightarrow φ = \frac{\sin (\frac{6 π}{7})}{2 \sin (\frac{π}{7})} = \frac{1}{2}

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