"If z= cis(ø) and $\displaystyle w = (1+z)/(1+z*)$ , show that w = cis ø"
Help! I have tried, but failed on this one!! It is high school level maths!
This is a complicated exercise in factoring.
First write $\displaystyle z=\cos(\phi)+i\sin(\phi)$.
Then note that $\displaystyle \frac{{1 + z}}{{1 + \overline z }} = \frac{{\left( {1 + z} \right)^2 }}{{\left| {1 + z} \right|^2 }}$.
But $\displaystyle \left| {1 + z} \right|^2 = \left( {1 + \cos (\phi )} \right)^2 + \sin ^2 (\phi ) = 2 + 2\cos (\phi )$.
Also $\displaystyle \left( {1 + z} \right)^2 = 1 + 2\cos (\phi ) + 2i\sin (\phi ) + \cos (2\phi ) + i\sin (2\phi ) = $$\displaystyle \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.