I have the following question.
Let $z_1$ and $z_2$ be complex numbers.
Assumptions:
-
$|z_1|=|z_2|=1$
-
$ z_1z_2 \neq -1$
What I have to prove is that:
$$\frac{z_1+z_2}{1+z_1z_2}$$
is real.
My thoughts:
First, I multiplied the numerator and denominator by the conjugate of the denominator. Therefore, the denominator is real for sure, because of the proof I know that:
$$z\bar z = |z|^2 $$
Therefore it is real, and I can ignore the denominator.
The result is now:
$$(z_1 + z_2)(1 + \overline{z_1}\,\overline{z_2})$$
Another thought I had is to use trigonometric identities, but that do very well either.
Any help is appreciated.
Best Answer
Hint: continue the expansion $$ (z_1+z_2)(1+\overline{z_1}\,\overline{z_2})= z_1+z_2+z_1\overline{z_1}\,\overline{z_2}+\overline{z_1}\,z_2\overline{z_2} $$