prove that if w,z are complex numbers such that |w| = |z| = 1 and wz does not equal -1, then (w+z)/(1+wz) is a real number
Easiest is to work with the polar form of the number.
The modulus of w is 1 so in polar form you can let it equal cos(A) + i.sin(A).
Similarly you can let z = cos(B) + i.sin(B).
Substitute those in and carry out the addition on the top line and the multiplication on the bottom line.
Next is to multiply top and bottom by the conjugate of the cn on the bottom line.
Picking out the imaginary part of the top line, (there's no need to bother with the rest of it), you should find that it is
-(cosA + cosB)(sinA.cosB + cosA.sinB) + (1 + cosA.cosB - sinA.sinB )(sinA + sinB),
(multiplied by i that is).
Multiply that out and tidy up (you'll need to make use of the trig identity cos^2 + sin^2 = 1, (twice)), and you should find that it reduces to zero, implying that the fraction is real.