Proving a result

substitute x = sin A and y = sin B (There is a small rider, we will chose A and B such that cosA and cosB are positive, otherwise this solution will fail!) Did u figure out why??!!
The equations get reduced to...
cos A + cos B = a (sin A - sin B)
2 cos (A+B)/2 . cos (A-B)/2 = 2 a cos (A+B)/2 . sin (A-B)/2
cos (A-B)/2 = a sin (A-B)/2 ........(1)
Thus, cot (A-B)/2 = a
A - B = 2 cot^-1a
or sin^-1x - sin^-1y - 2 cot^-1a
Differentiating above eq. w.r.t.x
1/√(1-x²) - 1/√(1-y²) dy / dx = 0
dy/dx=√(1-y²)/(1-x²)