Question Number 39028 by maxmathsup by imad last updated on 01/Jul/18
$$\left.\mathrm{1}\right)\:{calculate}\:\:{A}={cos}\left(\frac{\pi}{\mathrm{7}}\right).{cos}\left(\frac{\mathrm{2}\pi}{\mathrm{7}}\right).{cos}\left(\frac{\mathrm{3}\pi}{\mathrm{7}}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{B}\:={tan}\left(\frac{\pi}{\mathrm{7}}\right).{tan}\left(\frac{\mathrm{2}\pi}{\mathrm{7}}\right).{tan}\left(\frac{\mathrm{3}\pi}{\mathrm{7}}\right). \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 02/Jul/18
$${A}×\mathrm{2}{sin}\frac{\Pi}{\mathrm{7}}=\mathrm{2}{sin}\frac{\Pi}{\mathrm{7}}{cos}\frac{\Pi}{\mathrm{7}}{cos}\frac{\mathrm{2}\Pi}{\mathrm{7}}{cos}\frac{\mathrm{3}\Pi}{\mathrm{7}} \\ $$$${A}×\mathrm{2}{sin}\frac{\Pi}{\mathrm{7}}={sin}\frac{\mathrm{2}\Pi}{\mathrm{7}}{cos}\frac{\mathrm{2}\Pi}{\mathrm{7}}{cos}\frac{\mathrm{3}\Pi}{\mathrm{7}} \\ $$$${A}×\mathrm{2}^{\mathrm{2}} {sin}\frac{\Pi}{\mathrm{7}}={sin}\frac{\mathrm{4}\Pi}{\mathrm{7}}{cos}\frac{\mathrm{3}\Pi}{\mathrm{7}} \\ $$$${A}×\mathrm{2}^{\mathrm{3}} {sin}\frac{\Pi}{\mathrm{7}}=\mathrm{2}{sin}\frac{\mathrm{4}\Pi}{\mathrm{7}}{cos}\frac{\mathrm{3}\Pi}{\mathrm{7}} \\ $$$${A}×\mathrm{2}^{\mathrm{3}} {sin}\frac{\Pi}{\mathrm{7}}={sin}\Pi+{sin}\frac{\Pi}{\mathrm{7}} \\ $$$${A}×\mathrm{2}^{\mathrm{3}} {sin}\frac{\Pi}{\mathrm{7}}={sin}\frac{\Pi}{\mathrm{7}} \\ $$$${A}=\frac{\mathrm{1}}{\mathrm{8}} \\ $$