15. (i) Prove that cos 40 - cos 4a ...
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15. (i) Prove that cos 40 - cos 4a = 8(cos (cos e-sin a)(cos + sin a) - cos a)(cos + cosa)

11th - 12th Class
Maths
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( cos 4 theta-cos 4 alpha quad cos ^{2} theta=pm cos 2 theta ) ( =left(2 cos ^{2} 2 theta-1right)-left(2 cos ^{2} 2 alpha-1right) quad cos ^{2} 2 theta equiv frac{1+cos 4 theta}{2} ) ( =2 cos ^{2} 2 theta-2 cos ^{2} 2 alpha-1+1 ) ( =2left[cos ^{2} 2 theta-cos ^{2} 2 alpharight] ldots quad cos alpha theta=2 cos ^{2} 2 theta-1 ) ( =2(cos 2 theta+cos 2 alpha)(cos 2 theta-cos 2 alpha)^{2}-b^{2}=(a+b)(9-b) ) ( 8=2left(2 cos ^{2} theta-1+2 cos ^{2} alpha-1right)left(2 cos ^{2} theta-1-left(2 cos ^{2} alpha-1right)right) ) ( =2left[2left(cos ^{2} theta+cos ^{2} alpha-1right)left(2 cos ^{2} theta-1-2 cos ^{2} alpha+alpharight)right. ) ( left.=2left[2 cos ^{2} theta+cos ^{2} alpha-1right) quad 2left(cos 2 theta-cos ^{2} alpharight)right] ) ( =8 quadleft[cos ^{2} theta+cos ^{2} alpha-1right) quadleft(cos ^{2} theta-cos ^{2} alpharight) ) ( =8left(cos ^{2} theta-cos ^{2} alpharight)left(cos ^{2} theta+cos ^{2} alpha-1right) quad int sin ^{2} alpha+cos ^{2} alpha ) ( =8left(cos ^{2} theta-cos ^{2} alpharight)left(cos ^{2} theta+cos ^{2} alpha-left(sin ^{2} alpha+cos ^{2} alpharight)right) quad=1 ) ( =8left(cos ^{2} theta-cos ^{2} alpharight)left[cos ^{2} theta+cos ^{2} alpha-sin ^{2} alpha-cos ^{2} alpharight) ) ( =frac{8left(cos ^{2} theta-cos ^{2} alpharight)}{2} frac{left.cos ^{2} theta-sin ^{2} alpharight]}{L} cdot 9^{2}-b^{2}=(4+b)(a-b) ) ( =cdot 0(cos theta+cos alpha)(cos theta-cos 2)(cos theta+sin alpha) cos theta-sin alpha) ) ( = ) LHS = RItS proved.
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