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cos20-cos40-cos-2-80-




Question Number 200823 by mathlove last updated on 24/Nov/23
cos20∙cos40+cos^2 80=?
$${cos}\mathrm{20}\centerdot{cos}\mathrm{40}+{cos}^{\mathrm{2}} \mathrm{80}=? \\ $$
Answered by som(math1967) last updated on 24/Nov/23
(1/2)(2cos20cos40+2cos^2 80)  =(1/2)(cos20+cos60+1+cos160)  =(1/2)(cos20+(1/2)+1−cos20)  =(1/2)×(3/2)=(3/4)
$$\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{2}{cos}\mathrm{20}{cos}\mathrm{40}+\mathrm{2}{cos}^{\mathrm{2}} \mathrm{80}\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left({cos}\mathrm{20}+{cos}\mathrm{60}+\mathrm{1}+{cos}\mathrm{160}\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left({cos}\mathrm{20}+\frac{\mathrm{1}}{\mathrm{2}}+\mathrm{1}−{cos}\mathrm{20}\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}×\frac{\mathrm{3}}{\mathrm{2}}=\frac{\mathrm{3}}{\mathrm{4}} \\ $$

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