How do i prove that 2 cot 2x = cot x-tan x?

We will start off working on the right hand side. 
cot x - tan x 
= [cos x / sin x] - [sin x / cos x] 
= [(cos x)^ 2 - (sin x)^2] / [sin x cos x] 

This is where it gets a bit tougher if you do not have your formula list with you. 
(cos x)^ 2 - (sin x)^2 = cos(2x) 
sin 2x = 2 sin x cos x 

Note that by arranging the second formula, we will have sin x cos x = (1/2) sin 2x 

Hence, we will get: 
[(cos x)^ 2 - (sin x)^2] / [sin x cos x] 
= [cos 2x] / (1/2)[sin 2x] 
= 2[cos 2x] / [sin 2x] 
= 2cot 2x 

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