1980 TYS

If $z=\cos \theta +i\sin \theta $, prove that $\frac{1+z}{1-z}=i\cot \frac{\theta }{2}$ . In an Argand diagram, the points $O$, $A$, $Z$, $P$, $Q$ represent the complex numbers $0$, $1$, $z$, $1+z$ and $1-z$ respectively. Indicate these points on a diagram, showing clearly their geometrical relationships. Prove that angle $POQ=\frac{\pi }{2}$ and find, in terms of $\theta $, the ratio $\left| \frac{OP}{OQ} \right|$.