If $\displaystyle z = cis\theta$, where $\displaystyle \theta < \frac{\pi}{4} $ show that:

$\displaystyle \frac{1-z^2}{z+1} = 2\sin(\frac{\theta}{2})cis(\pi-\frac{3\theta}{2})$

Any help would be much appreciated as all my attempts at solving this haven't gone too well.