Operations with complex numbers to give real numbers

Question

If:

1. $|z|=|w|=1$

2. $1 + zw \neq 0$

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

Answer 1

Hint: Take the complex conjugate then multiply top and bottom by $zw$.

Answer 2

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.