CBSE Class 12-science Answered

Given  cos(x-y), cosx, cos(x+y). are in HP

So, 1/cos(x-y), 1/cos x, 1/cos (x+y) are in AP

Therefore,

2/ cos x = 1/ cos(x-y) + 1/cos (x+y)

2/cosx = [cos (x+y) + cos (x-y) ] / [cos(x-y) . cos (x+y)]

2/ cosx = 2 cos(x) cos (y) / cos²x - sin²y

1/ cosx = cos(x) cos (y) / cos²x - sin²y

By cross multiplication,

cos² x . cosy = cos²x - sin²y

cos²x - cos²x . cosy = sin²y = 1 - cos²y

cos²x [1- cosy] = (1 - cosy)(1+ cosy)

cos²x = 1 + cosy

cos²x = 2 cos²(y/2)

cos²x/ cos²(y/2) = 2

cos²x sec²(y/2) = 2

cosx sec(y/2) = +sqrt(2) or -sqrt(2)

|cosx sec(y/2)| = sqrt(2)