Find the value of \[ tan\frac{1}{2} \bigg[ sin^{-1} \frac{2x}{1+x^2} + cos^{-1} \frac{1-y^2}{1+y^2} \bigg], | x | < 1, y > 0 \: and \: xy < 1\]

Question

$\begin{array}{1 1} (\large \frac{x+y}{1-xy} ) \\ (\large \frac{x-y}{1+xy} ) \\ (\large \frac{1+xy}{x-y} ) \\ (\large \frac{1-xy}{x+y} ) \end{array} $