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(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]