1. Complex numbers question

"If z= cis(ø) and $w = (1+z)/(1+z*)$ , show that w = cis ø"

Help! I have tried, but failed on this one!! It is high school level maths!

2. Originally Posted by karldiesen
"If z= cis(ø) and $w = (1+z)/(1+z*)$ , show that w = cis ø"
This is a complicated exercise in factoring.
First write $z=\cos(\phi)+i\sin(\phi)$.
Then note that $\frac{{1 + z}}{{1 + \overline z }} = \frac{{\left( {1 + z} \right)^2 }}{{\left| {1 + z} \right|^2 }}$.

But $\left| {1 + z} \right|^2 = \left( {1 + \cos (\phi )} \right)^2 + \sin ^2 (\phi ) = 2 + 2\cos (\phi )$.

Also $\left( {1 + z} \right)^2 = 1 + 2\cos (\phi ) + 2i\sin (\phi ) + \cos (2\phi ) + i\sin (2\phi ) =$ $\cos (\phi )\left[ {2 + 2\cos (\phi )} \right] + i\sin (\phi )\left[ {2 + 2\cos (\phi )} \right].$

I will let you have the fun of doing the details.