m taking y in place of alpha, nd x in place of beta,
tan2y = cos2x - sin2x ....................(1)
subtracting 1 frm both sides, we have
tan2y -1 = cos2x - sin2x -1
=> cos2y - sin2y = 2 sin2x cos2y ....................(2)
now adding 1 to the equation 1,
tan2y +1 = cos2x - sin2x +1
=> sec2y = 2 cos2x
=> cos2y = 1/ (2 cos2x )
thrfore, eqn 2 will be now
cos2y - sin2y = 2 sin2x / 2 cos2x
cos2y - sin2y = tan2x
hence proved