Question

Find the value of the following:
$\tan \frac{1}{2}\left[\begin{array}{c}{\sin }^{-1}\frac{2x}{1+x^2}+{\cos }^{-1}\frac{1-y^2}{1+y^2}\end{array}\right],|x|<1,y<0$ and $xy<1$.

Answer 1

$\tan \left\{\frac{1}{2}{\sin }^{-1}(\frac{2x}{1+x^2}+\frac{1}{2}{\cos }^{-1}\left(\frac{1-y^2}{1+y^2}\right))\right\}$

$=\tan \left\{\frac{1}{2}\right(2{\tan }^{-1}x)+\frac{1}{2}(2{\tan }^{-1}y\left)\right\}$

$=\tan ({\tan }^{-1}x+{\tan }^{-1}y)$

$=\frac{x+y}{1-xy}$