Question

Prove that (sin3A + sinA)sinA + (cos3A-COSA) COSA = 0

JEE/Engineering Exams

Maths

Solution

127

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E P.T ( (operatorname{lin} 3 A+sin A) sin A+(cos 3 A-cos A) cos A=0 ) ( Rightarrow sin 3 A=3 sin A-4 sin ^{3} A quad 1 quad cos 3 A=4 cos ^{3} A-3 cos A ) ( Rightarrow quad(sin 3 A+sin A) sin A+(cos 7 A-cos A) cos A ) ( Rightarrowleft(3 sin A-4 sin ^{3} A+sin Aright) sin A+(4 cos 3 A-3 cos A-cos A) cos A ) ( Rightarrowleft(4 sin A-4 sin ^{3} Aright) sin A+left(4 cos ^{3} A-4 cos Aright) cos A ) ( Rightarrow quad 4 sin ^{2} Aleft(1-sin ^{2} Aright)+4 cos ^{2} Aleft(cos ^{2} A-1right) ) ( Rightarrow quad 4 sin ^{2} A cdot cos ^{2} A cdot frac{pi}{6} cdot 4 cos ^{2} A cdot sin ^{2} A ) ( Rightarrow quad 0 )