Let $f\left(x\right)={\sin }^{-1}\left(\frac{2x}{1+x^2}\right)andg\left(x\right)={\cos }^{-1}\left(\frac{x^2-1}{x^2+1}\right)$. Then the value of $f\left(10\right)-g\left(100\right)$ is equal to

$f\left(x\right)={\sin }^{-1}\left(\frac{2x}{1+x^2}\right)=\pi -2{\tan }^{-1}x,forx\ge 1$ and

$g\left(x\right)={\cos }^{-1}\left(\frac{x^2-1}{x^2+1}\right)=\pi -{\cos }^{-1}\left(\frac{1-x^2}{1+x^2}\right)$

$=\pi -2{\tan }^{-1}x,forx\ge 0$

Now, $f\left(10\right)-g\left(100\right)$

$=(\pi -2{\tan }^{-1}\left(10\right))-(\pi -2{\tan }^{-1}\left(100\right))$

$=2({\tan }^{-1}\left(100\right)-{\tan }^{-1}\left(10\right))$