: this question has been driving me nuts for days!!
it says to prove that:
(1+z)/(1+conjugate z) = z 
                     
This is true if and only if |z| = 1.  Are you sure you
weren't given that |z| = 1?

You can't prove it in general because it isn't true in
general! You can easily DISprove that it is true in 
general with a counterexample.

Suppose z = 1+i
     _
Then z = 1-i

Then 
         _  
(1+z)/(1+z) =

 1 + (1+i)     2 + i   2 + i     4 + 4i + i²
----------- = -------·------- = ------------- =
 1 + (1-i)     2 - i   2 + i        4 - i²


4 + 4i + (-1)     3 + 4i
------------- =  -------- = (3/5) + (4/5)i
   4 - (-1)         5

which does not equal z, since z = 1+i

=================================================

However if you were given that |z| = 1, and I think
you were, then it is true.

Proof:
                                       _
First we must prove the lemma: |z|² = zz

Suppose z = x+yi where x and y are real numbers
       _____
|z| = Öx²+y², so |z|² = x²+y²
 _
zz = (x+yi)(x-iy) = x²-y²i² = x²-y²(-1) = x²+y²

Therefore the lemma is true.

     |z| = 1 is given

Square both sides

    |z|² = 1
                 _
Replace |z|² by zz, using the lemma 
       _
      zz = 1

Add z to both sides
       _
  z + zz = 1 + z

Factor out z on the left
      _
z(1 + z) = 1 + z
                          _
Divide both sides by (1 + z)
                        _ 
       z = (1 + z)/(1 + z)

QED

Edwin