Rotation in complex plane

If, for complex z and w, , show that and differ by .

I have an idea how to solve this. Drive out from the initial equation (where ), then do something with rotating tangents around 90 degrees blahblah. The problem is I don't really know how to put this down on paper as I get kinda lost when it comes to these funky substitutions into arctan functions. Or maybe there is another way to do this?

(We didn't do vectors yet, so I should perhaps do without them, but any hint you can drop on dot and cross products and whatnot will be appreciated.) Thanks!