Question

prove that cos^2 15-cos^2 75=root 3 byb2
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Answer 1

cos^2 - cos^75 =(cos15-cos75) (cos15+cos75)

we know that

cosx-cosy = -2sin(X+y/2)sin(x-y/2)

cosx+cosy = 2cos(x+y/2)cos(x-y/2)

so, [-2sin(15+75/2)sin(15-75/2)] [ 2cos(15+75/2) cos(15-75/2)]

[-2sin45°sin(-30)] [2cos45cos30]

[2sin45sin30] [2cos45cos30]

[2×1/√2×1/2] [2×1/√2×√3/2]

1/√2 × √3/√2

√3/2

hence proved