# prove that $Z_1 = i\cot \frac{\theta}{2}$ given that $Z = \cos \theta + i\sin \theta$

If $Z = \cos \theta+i\sin \theta$, $Z_1= \dfrac{z+1}{z-1}$, prove that $Z_1= -i\cot \dfrac{θ}{2}\\$

This was a proof that I ran into in a quiz. I couldn't really solve it. I only got as far as $Z_1 = z+1$, $Z_1 = \cos \theta+\sin \theta$ $-1$.

It's bugging me since then. A friend told me we'd prove it using a double angle formula.

• write $Z$ as $e^{i\theta}$ and factor by $e^{i\frac{\theta}{2}}$. – hamam_Abdallah Oct 21 '16 at 13:43

• I put a negative in front of the $i$ in the OP – imranfat Oct 21 '16 at 15:24