Q:

Find the value of the real part of (z+w)/(z-w) when |z| = |w|

It's an IBDP question. 
When I try to solve it, I end up with a fraction with many variables. 
I was thinking maybe I should rationalize the fraction...

A:

Write $z=re^{it}$ and $w=re^{iu}$. Then
$$\frac{z+w}{z-w}=\frac{e^{it}+e^{iu}}{e^{it}-e^{iu}}
=\frac{e^{i(t-u)/2}+e^{-i(t-u)/2}}{e^{i(t-u)/2}-e^{-i(t-u)/2}}
=\frac{2\cos\frac12(t-u)}{2i\sin\frac12(t-u)}=i\cot\theta$$
where $\theta=\frac12(u-t)$ is half the angle between $z$ and $w$ in the
Argand diagram. So zero real part, but the imaginary part is interesting.

