Question Number 29833 by abdo imad last updated on 12/Feb/18
$${find}\:\:{cos}^{\mathrm{4}} \left(\frac{\pi}{\mathrm{8}}\right)\:+{cos}^{\mathrm{4}} \left(\frac{\mathrm{3}\pi}{\mathrm{8}}\right)\:+{cos}^{\mathrm{4}} \left(\frac{\mathrm{5}\pi}{\mathrm{8}}\right)\:+{cos}^{\mathrm{4}} \left(\frac{\mathrm{7}\pi}{\mathrm{8}}\right). \\ $$
Answered by MJS last updated on 14/Feb/18
$$\mathrm{cos}\left(\frac{\pi}{\mathrm{8}}\right)=−\mathrm{cos}\left(\frac{\mathrm{7}\pi}{\mathrm{8}}\right)=\frac{\sqrt{\mathrm{2}+\sqrt{\mathrm{2}}}}{\mathrm{2}} \\ $$$$\mathrm{cos}\left(\frac{\mathrm{3}\pi}{\mathrm{8}}\right)=−\mathrm{cos}\left(\frac{\mathrm{5}\pi}{\mathrm{8}}\right)=\frac{\sqrt{\mathrm{2}−\sqrt{\mathrm{2}}}}{\mathrm{2}} \\ $$$$\mathrm{so}\:\mathrm{the}\:\mathrm{answer}\:\mathrm{is}\:\frac{\mathrm{3}}{\mathrm{2}} \\ $$