Q:

Prove $\ \frac{z^{2010} - \bar z^{2010}}{1+z\bar z}$ is imaginary number

Prove $\ \frac{z^{2010} - \bar z^{2010}}{1+z\bar z}$ is imaginary number. 
I understand that if
$\ z = (a+bi) $ then $\ z - \bar z = 2bi  $ and the denominator $\ 1+z\bar z $ is $\ 1+|z|^2 $ and therefore it is a real number.  so need to prove the numerator is imaginary. 
I first tried to see what happens if I take $\ z^2 - \bar z^2 $ and it is imaginary $\ (a+bi)^2-(a-bi)^2 = 4abi $
and also $ z^{2010} = (z^{1005})^2$ so if $\ z^{1005} $ is imaginary number... but I have no clue  what is $\ z^{1005}$ and maybe it's a real number..? 

A:

It’s much simpler than it might look at first glance. 
First step: No matter what the (nonreal) complex number $z=a+bi$ is, $z-\bar z$ is purely imaginary, namely equal to $2bi$.
Second step: $(\bar z)^m=\overline{(z^m)}$.
Third step: no matter what the complex number $z$ is, we get $z\bar z$ to be real and nonnegative, equal to $a^2+b^2$.
Put them all together and see that your fraction is a purely imaginary number divided by a positive real number.