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
