2 cot 2x=cot x-tan x

Often with problems like these it is helpful to convert everything to sin and cos.

Then you get from the identity tan(x)=\frac{sin(x)}{cos(x)}

\frac{2cos(2x)}{sin(2x)}=\frac{cos(x)}{sin(x)}-\frac{sin(x)}{cos(x)}

Now, cos(2x)=cos^{2}(x)-sin^{2}(x), \;\ sin(2x)=2sin(x)cos(x):

So, we have:

\frac{2(cos^{2}(x)-sin^{2}(x))}{2sin(x)cos(x)}=\frac{cos^{2}(x)-sin^{2}(x)}{sin(x)cos(x)}

Which equals the right side.