(cos x)/(sec x - 1) - (cos x)/(sec x + 1) = 2 cos^3 x csc^2 x
Get a common denominator on the left, Shades....let's see what happens
[cosx(secx + 1) - cosx(sec - 1)] / [ (secx - 1) (secx + 1) ]
[cosxsecx + cosx - cosxsecx + cosx] / [tan^2x] =
[ 2cosx] / [tan^2x] =
[2cosx[ / [ sin^2x/ cos^2x] =
[2cosx] * [cos^2x /sin^2x] =
[2cos^3x] * [ 1 / sin^2x] =
[2cos^3x] [csc^2x]