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..?

Using trigonometric form:
$$z^{2010}-\bar{z}^{2010}=r^{2010}(\cos (2010t)+i\sin (2010t))-r^{2010}(\cos(2010t)-i\sin(2010t))=\\
2r^{2010}\sin(2010t)\cdot i.$$
