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Question Number 29833 by abdo imad last updated on 12/Feb/18

find  cos^4 ((π/8)) +cos^4 (((3π)/8)) +cos^4 (((5π)/8)) +cos^4 (((7π)/8)).

$${find}\:\:{cos}^{\mathrm{4}} \left(\frac{\pi}{\mathrm{8}}\right)\:+{cos}^{\mathrm{4}} \left(\frac{\mathrm{3}\pi}{\mathrm{8}}\right)\:+{cos}^{\mathrm{4}} \left(\frac{\mathrm{5}\pi}{\mathrm{8}}\right)\:+{cos}^{\mathrm{4}} \left(\frac{\mathrm{7}\pi}{\mathrm{8}}\right). \\ $$

Answered by MJS last updated on 14/Feb/18

cos((π/8))=−cos(((7π)/8))=((√(2+(√2)))/2)  cos(((3π)/8))=−cos(((5π)/8))=((√(2−(√2)))/2)  so the answer is (3/2)

$$\mathrm{cos}\left(\frac{\pi}{\mathrm{8}}\right)=−\mathrm{cos}\left(\frac{\mathrm{7}\pi}{\mathrm{8}}\right)=\frac{\sqrt{\mathrm{2}+\sqrt{\mathrm{2}}}}{\mathrm{2}} \\ $$$$\mathrm{cos}\left(\frac{\mathrm{3}\pi}{\mathrm{8}}\right)=−\mathrm{cos}\left(\frac{\mathrm{5}\pi}{\mathrm{8}}\right)=\frac{\sqrt{\mathrm{2}−\sqrt{\mathrm{2}}}}{\mathrm{2}} \\ $$$$\mathrm{so}\:\mathrm{the}\:\mathrm{answer}\:\mathrm{is}\:\frac{\mathrm{3}}{\mathrm{2}} \\ $$

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