N1980 P1 Q4

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 \(\left| \frac{OP}{OQ} \right|\).