If $|z|<1$ , show that $\left|\frac{1}{2}\arg (\frac{1+z}{1-z}) \right| < \frac{\pi}{2}$

Question

If $|z|<1$ , show that $\left|\frac{1}{2}\arg (\frac{1+z}{1-z}) \right| < \frac{\pi}{2}$

Answer 1

$$\frac{1+x+iy}{1-x-iy}=\frac{(1+x+iy)(1-x+iy)}{(1-x)^2+y^2}=\frac{1-y^2-x^2+2y i}{(1-x)^2+y^2}$$

From the definition of atan2, here we need $1-y^2-x^2>0$