Calculate-cos-7-cos-4-7-cos-5-7-

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

$$\boldsymbol{\mathrm{Calculate}}. \\ $$$$\boldsymbol{\mathrm{cos}}\frac{\boldsymbol{\pi}}{\mathrm{7}}×\boldsymbol{\mathrm{cos}}\frac{\mathrm{4}\boldsymbol{\pi}}{\mathrm{7}}×\boldsymbol{\mathrm{cos}}\frac{\mathrm{5}\boldsymbol{\pi}}{\mathrm{7}}. \\ $$

Answered by ajfour last updated on 04/Apr/17

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

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