One way to do it is just by a direct approach, knowing
e^iθ=cosθ+isinθ:
q
=(cosθ+isinθ)/(1-cosθ-isinθ)
=[(cosθ+isinθ)(1-cosθ+isinθ)]/[(1-cosθ-isinθ)(1-cosθ+isinθ)]
=[(cosθ+isinθ-(cosθ)^2-isinθcosθ+isinθcosθ-sin^θ]/[(1-cosθ)^2+(sinθ)^2]
=(cosθ+isinθ-1)/[1-2cosθ+(cosθ)^2+(sinθ)^2]
=(cosθ-1)/(2-2cosθ)+i[sinθ/(2-2cosθ)]
therefore, the real part is
(cosθ-1)/(2-2cosθ)=(cosθ-1)/2(1-cosθ)=-1/2
There is a short-cut to this, I will write after school if you
need it.