Operations with complex numbers to give real numbers If:


*

*$|z|=|w|=1$

*$1 + zw \neq 0$
Then $\dfrac{z+w}{1+zw}$ is real.  How can prove that.

Let $z=e^{ia}$ and $w=e^{ib}$ for some $a,b \in [0,2\pi)$. This takes care of condition 1. Then $1+zw \neq 0$ means $1+e^{i(a+b)} \neq 0$, which is the same as saying $a+b$ is not an odd multiple of $\pi$.
Now consider
\begin{align*}
 \dfrac{z+w}{1+zw} & = \frac{e^{ia}+e^{ib}}{1+e^{i(a+b)}}
\end{align*}
Rationalize the denominator and see what happens.  
