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Question Number 129234 by bemath last updated on 14/Jan/21
 cos^2 47°+cos^2   73°+cos 47°cos 73°+(1/2)=?
$$\:\mathrm{cos}\:^{\mathrm{2}} \mathrm{47}°+\mathrm{cos}^{\mathrm{2}} \:\:\mathrm{73}°+\mathrm{cos}\:\mathrm{47}°\mathrm{cos}\:\mathrm{73}°+\frac{\mathrm{1}}{\mathrm{2}}=? \\ $$
Answered by liberty last updated on 14/Jan/21
(cos 73°+cos 47°)^2 −cos 73°cos 47°+(1/2)=  (2cos 60°cos 13°)^2 −(1/2)(cos 120°+cos 26°)+(1/2)=   cos^2 13°−(1/2)(−(1/2)+2cos^2 13°−1)+(1/2)=   cos^2 13°+(3/4)−cos^2 13°+(1/2)=(5/4)
$$\left(\mathrm{cos}\:\mathrm{73}°+\mathrm{cos}\:\mathrm{47}°\right)^{\mathrm{2}} −\mathrm{cos}\:\mathrm{73}°\mathrm{cos}\:\mathrm{47}°+\frac{\mathrm{1}}{\mathrm{2}}= \\ $$$$\left(\mathrm{2cos}\:\mathrm{60}°\mathrm{cos}\:\mathrm{13}°\right)^{\mathrm{2}} −\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{cos}\:\mathrm{120}°+\mathrm{cos}\:\mathrm{26}°\right)+\frac{\mathrm{1}}{\mathrm{2}}= \\ $$$$\:\mathrm{cos}\:^{\mathrm{2}} \mathrm{13}°−\frac{\mathrm{1}}{\mathrm{2}}\left(−\frac{\mathrm{1}}{\mathrm{2}}+\mathrm{2cos}\:^{\mathrm{2}} \mathrm{13}°−\mathrm{1}\right)+\frac{\mathrm{1}}{\mathrm{2}}= \\ $$$$\:\mathrm{cos}\:^{\mathrm{2}} \mathrm{13}°+\frac{\mathrm{3}}{\mathrm{4}}−\mathrm{cos}\:^{\mathrm{2}} \mathrm{13}°+\frac{\mathrm{1}}{\mathrm{2}}=\frac{\mathrm{5}}{\mathrm{4}} \\ $$

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