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Question Number 11900 by @ANTARES_VY last updated on 04/Apr/17

$$\mathbf{Calculate}.$$$$\mathbf{cos}\frac{\mathit{\pi }}{7}\times \mathbf{cos}\frac{4\mathit{\pi }}{7}\times \mathbf{cos}\frac{5\mathit{\pi }}{7}.$$

Answered by ajfour last updated on 04/Apr/17

$$=\cos \frac{\pi }{7}\cos \frac{4\pi }{7}[-\cos (\pi -\frac{5\pi }{7})]$$$$=-\cos \frac{\pi }{7}\cos \frac{4\pi }{7}\cos \frac{2\pi }{7}$$$$=\frac{-1}{2\sin \frac{\pi }{7}}\left[2\sin \frac{\pi }{7}\cos \frac{\pi }{7}\right]\left[\cos \frac{2\pi }{7}\cos \frac{4\pi }{7}\right]$$$$=-\frac{1}{2\sin \frac{\pi }{7}}\left[\sin \frac{2\pi }{7}\cos \frac{2\pi }{7}\right]\cos \frac{4\pi }{7}$$$$=-\frac{1}{4\sin \frac{\pi }{7}}\left(\sin \frac{4\pi }{7}\cos \frac{4\pi }{7}\right)$$$$=-\left(\frac{\sin \frac{8\pi }{7}}{8\sin \frac{\pi }{7}}\right)=-\frac{\sin (\pi +\frac{\pi }{7})}{8\sin \frac{\pi }{7}}$$$$=-\frac{-\sin \left(\pi /7\right)}{8\sin \left(\pi /7\right)}=\frac{1}{8}.$$

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