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Question-123385




Question Number 123385 by help last updated on 25/Nov/20
Answered by Dwaipayan Shikari last updated on 25/Nov/20
sin^6 1°+cos^6 1°+sin^6 2°+cos^6 2°+...+sin^6 n°+cos^6 n°  (1−3sin^2 1°cos^2 1°+1−3sin^2 2°cos^2 2°+...)+1  =45−(3/4)(sin^2 2°+sin^2 4°+...+sin^2 90°)  =45−(3/8)(1−cos4°+1−cos8°+....)  =((225)/8)+(3/8)(Σ_(n=1) ^(45) cos4n°)=((222)/8)=((111)/4)=27.75  Σ_(n=1) ^n cosan°= cosa°+cos2a°+..  =(1/(2sin2°))(sin(a+2)°−sin(a−2)°+...+sin(na+2)−sin(na−2))  =(1/(2sin2°))(sin(na+2)−sin(a−2))  =(1/(sin2°))(cos((n+1)(a/2))sin((n−1)(a/2)+2)  a=4° n=45  =−1
$${sin}^{\mathrm{6}} \mathrm{1}°+{cos}^{\mathrm{6}} \mathrm{1}°+{sin}^{\mathrm{6}} \mathrm{2}°+{cos}^{\mathrm{6}} \mathrm{2}°+…+{sin}^{\mathrm{6}} {n}°+{cos}^{\mathrm{6}} {n}° \\ $$$$\left(\mathrm{1}−\mathrm{3}{sin}^{\mathrm{2}} \mathrm{1}°{cos}^{\mathrm{2}} \mathrm{1}°+\mathrm{1}−\mathrm{3}{sin}^{\mathrm{2}} \mathrm{2}°{cos}^{\mathrm{2}} \mathrm{2}°+…\right)+\mathrm{1} \\ $$$$=\mathrm{45}−\frac{\mathrm{3}}{\mathrm{4}}\left({sin}^{\mathrm{2}} \mathrm{2}°+{sin}^{\mathrm{2}} \mathrm{4}°+…+{sin}^{\mathrm{2}} \mathrm{90}°\right) \\ $$$$=\mathrm{45}−\frac{\mathrm{3}}{\mathrm{8}}\left(\mathrm{1}−{cos}\mathrm{4}°+\mathrm{1}−{cos}\mathrm{8}°+….\right) \\ $$$$=\frac{\mathrm{225}}{\mathrm{8}}+\frac{\mathrm{3}}{\mathrm{8}}\left(\underset{{n}=\mathrm{1}} {\overset{\mathrm{45}} {\sum}}{cos}\mathrm{4}{n}°\right)=\frac{\mathrm{222}}{\mathrm{8}}=\frac{\mathrm{111}}{\mathrm{4}}=\mathrm{27}.\mathrm{75} \\ $$$$\underset{{n}=\mathrm{1}} {\overset{{n}} {\sum}}{cosan}°=\:{cosa}°+{cos}\mathrm{2}{a}°+.. \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}{sin}\mathrm{2}°}\left({sin}\left({a}+\mathrm{2}\right)°−{sin}\left({a}−\mathrm{2}\right)°+…+{sin}\left({na}+\mathrm{2}\right)−{sin}\left({na}−\mathrm{2}\right)\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}{sin}\mathrm{2}°}\left({sin}\left({na}+\mathrm{2}\right)−{sin}\left({a}−\mathrm{2}\right)\right) \\ $$$$=\frac{\mathrm{1}}{{sin}\mathrm{2}°}\left({cos}\left(\left({n}+\mathrm{1}\right)\frac{{a}}{\mathrm{2}}\right){sin}\left(\left({n}−\mathrm{1}\right)\frac{{a}}{\mathrm{2}}+\mathrm{2}\right)\right. \\ $$$${a}=\mathrm{4}°\:{n}=\mathrm{45} \\ $$$$=−\mathrm{1} \\ $$$$ \\ $$$$ \\ $$

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