23. Prove that cos® A - sinº A = co...
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23. Prove that cos® A - sinº A = cos 2

JEE/Engineering Exams
Maths
Solution
152
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( cos ^{6} A-sin ^{6} A=cos 2 A /left(-frac{1}{4} sin ^{2} 2 Aright) ) ( frac{cos ^{6} A-sin ^{6} A}{2}=frac{sin ^{2}(cos A)}{cos ^{2} A^{2}+left(cos ^{2} A-sin ^{2} Aright)left(cos ^{2} Aright)^{2}+cos ^{2} A sin ^{2} A+left(sin ^{2} Aright)^{2}} ) ( =cos 2 Aleft(cos ^{2} A+sin ^{2} Aright)-cos ^{2} A sin ^{2} A ) ( =left(1-frac{1}{4}(2 cos A sin A)^{2}right) ) ( =cos 2 Aleft(1-frac{1}{4} sin ^{2} 2 Aright) ) ( =frac{1}{4} )
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