cos-2pi-21-cos-4pi-21-cos-8pi-21-cos-10pi-22-cos-16pi-21-cos-20pi-21-

Question Number 208252 by efronzo1 last updated on 09/Jun/24

cos (((2π)/(21)))cos (((4π)/(21)))cos (((8π)/(21)))cos (((10π)/(22)))cos (((16π)/(21)))cos (((20π)/(21)))=?

$$\:\mathrm{cos}\:\left(\frac{\mathrm{2}\pi}{\mathrm{21}}\right)\mathrm{cos}\:\left(\frac{\mathrm{4}\pi}{\mathrm{21}}\right)\mathrm{cos}\:\left(\frac{\mathrm{8}\pi}{\mathrm{21}}\right)\mathrm{cos}\:\left(\frac{\mathrm{10}\pi}{\mathrm{22}}\right)\mathrm{cos}\:\left(\frac{\mathrm{16}\pi}{\mathrm{21}}\right)\mathrm{cos}\:\left(\frac{\mathrm{20}\pi}{\mathrm{21}}\right)=? \\ $$

Answered by som(math1967) last updated on 09/Jun/24

let (π/(21))=x cos2xcos4xcos8xcos10xcos16xcos20x cosxcos2xcos4xcos8xcos10xcos5x [21x=π⇒cos20x=−cosx,cos16x=−cos5x] (1/(4sinxsin5x))×sin2xcos2xcos4xcos8xsin10xcos10x =(1/(64sinxsin5x))×2sin8xcos8x×2sin10xcoss10x =(1/(64sinxsin5x))×sin16xsin20x =(1/(64sinxsin5x))×sin5x×sinx [21x=π ⇒sin16x=sin5x,sin20x=sinx] =(1/(64))

$$\:{let}\:\frac{\pi}{\mathrm{21}}={x} \\ $$$${cos}\mathrm{2}{xcos}\mathrm{4}{xcos}\mathrm{8}{xcos}\mathrm{10}{xcos}\mathrm{16}{xcos}\mathrm{20}{x} \\ $$$${cosxcos}\mathrm{2}{xcos}\mathrm{4}{xcos}\mathrm{8}{xcos}\mathrm{10}{xcos}\mathrm{5}{x} \\ $$$$\left[\mathrm{21}{x}=\pi\Rightarrow{cos}\mathrm{20}{x}=−{cosx},{cos}\mathrm{16}{x}=−{cos}\mathrm{5}{x}\right] \\ $$$$\frac{\mathrm{1}}{\mathrm{4}{sinxsin}\mathrm{5}{x}}×{sin}\mathrm{2}{xcos}\mathrm{2}{xcos}\mathrm{4}{xcos}\mathrm{8}{xsin}\mathrm{10}{xcos}\mathrm{10}{x} \\ $$$$=\frac{\mathrm{1}}{\mathrm{64}{sinxsin}\mathrm{5}{x}}×\mathrm{2}{sin}\mathrm{8}{xcos}\mathrm{8}{x}×\mathrm{2}{sin}\mathrm{10}{xcoss}\mathrm{10}{x} \\ $$$$=\frac{\mathrm{1}}{\mathrm{64}{sinxsin}\mathrm{5}{x}}×{sin}\mathrm{16}{xsin}\mathrm{20}{x} \\ $$$$=\frac{\mathrm{1}}{\mathrm{64}{sinxsin}\mathrm{5}{x}}×{sin}\mathrm{5}{x}×{sinx} \\ $$$$\left[\mathrm{21}{x}=\pi\:\Rightarrow{sin}\mathrm{16}{x}={sin}\mathrm{5}{x},{sin}\mathrm{20}{x}={sinx}\right] \\ $$$$=\frac{\mathrm{1}}{\mathrm{64}} \\ $$

Commented by efronzo1 last updated on 09/Jun/24