cotx cot2x - cot2x cot3x-cot3x cotx=1
Here's the easiest method.
Find out, what's the biggest angle here? It's "cot 3x"
cot (3x) = cot ( 2x+x )
Apply, cot (A+B) = (cot A.cot B -1)/ cot A + cot B
cot (3x) = ( cot 2x.cot x - 1 ) / cot 2x + cot x
Now, cross multiply
cot 3x ( cot 2x + cot x ) = cot 2x.cot x - 1
cot 2x.cot 3x + cot x.cot 3x = cot 2x.cot x - 1
cot 2x.cot x - cot 2x.cot 3x - cot x.cot 3x = 1
This is by far the easiest method, I've came across.
This method is applicable for both tan and cot..! :)
Hope I helped..!! :D