There's one more way to solve it:

tan 3x - tan x = sin (3x- x) / (cos 3x cos x)

& cot 3x - cot x = sin(x - 3x) / (sin 3x sin x)

Putting in the given expression:

= cos 3x cos x / sin 2x - sin 3x sin x / sin (-2x)

= (cos 3x cos x + sin 3x sin x) / sin 2x

= cos 2x / sin 2x

= cot 2x.

This gives the same result.