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Question Number 168297 by MdNafiz last updated on 07/Apr/22

$$\mathrm{prove}\mathrm{that}........$$$$\cos \frac{2\mathrm{\Pi }}{7}+\cos \frac{4\mathrm{\Pi }}{7}+\cos \frac{8\mathrm{\Pi }}{7}=-\frac{1}{2}$$

Answered by som(math1967) last updated on 07/Apr/22

$$let\frac{\pi }{7}=\theta$$$$\therefore cos2\theta +cos4\theta +cos8\theta$$$$=\frac{2sin2\theta cos2\theta +2sin2\theta cos4\theta +2sin2\theta cos8\theta }{2sin2\theta }$$$$=\frac{sin4\theta +sin6\theta -sin2\theta +sin10\theta -sin6\theta }{2sin2\theta }$$$$=\frac{sin4\theta +sin10\theta -sin2\theta }{2sin2\theta }$$$$=\frac{2sin7\theta cos3\theta -sin2\theta }{2sin2\theta }$$$$=\frac{0-sin2\theta }{2sin2\theta }=-\frac{1}{2}★$$$$★sin7\theta =sin\pi =0$$

Commented by peter frank last updated on 08/Apr/22

$$\mathrm{good}$$

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