Question

Find the value of the following:

$tan\frac{1}{2}[si{n}^{-1}\frac{2x}{1+x^2}+co{s}^{-1}\frac{1-y^2}{1+y^2}],|x|<1,y>0andxy<1$

Answer 1

$tan\frac{1}{2}[si{n}^{-1}\frac{2x}{1+x^2}+co{s}^{-1}\frac{1-y^2}{1+y^2}][\because 2ta{n}^{-1}x=si{n}^{-1}\left(\frac{2x}{1+x^2}\right)and2ta{n}^{-1}y=co{s}^{-1}\left(\frac{1-y^2}{1+y^2}\right)]=tan\left[\frac{1}{2}\right(2ta{n}^{-1}x+2ta{n}^{-1}y\left)\right]=tan[\frac{1}{2}\times 2(ta{n}^{-1}x+ta{n}^{-1}y\left)\right]=tan(ta{n}^{-1}x+ta{n}^{-1}y)=tan\left[ta{n}^{-1}\left(\frac{x+y}{1-xy}\right)\right]=\frac{x+y}{1-xy}[\because ta{n}^{-1}x+ta{n}^{-1}y=ta{n}^{-1}\frac{x+y}{1-xy}]$