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!
Well, the above equality can be proved either by differentiating the LHS and checking it is zero and then evaluating at , or
even more basically by trigonometry: take a straight angle triangle with acute angles and opposite
legs of length , resp. Then and now just
apply to both sides and remember that