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

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

Commented by bemath last updated on 09/Jun/20

$$x=\frac{2\pi }{7}$$$$\frac{\cos x+\cos 2x+\cos 3x}{2\sin x}\times 2\sin x$$$$\frac{\sin 2x+2\sin x\cos 2x+2\sin x\cos 3x}{2\sin x}=$$$$\frac{\sin 2x+\sin 3x-\sin x+\sin 4x-\sin 2x}{2\sin x}=$$$$\frac{\sin 4x+\sin 3x-\sin x}{2\sin x}=$$$$\frac{2\sin \left(\frac{7x}{2}\right)\cos \left(\frac{x}{2}\right)-\sin x}{2\sin x}=-\frac{1}{2}$$$$[\sin \left(\frac{7x}{2}\right)=\sin \pi =0]$$

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