Find range of the function:$f(x)=(\sin^{-1}x)^2-(\cot^{-1}x)^2$
The domain of the function is $-1\leq x \leq 1$
$f(-1)=(\sin^{-1}(-1))^2-(\cot^{-1}(-1))^2=\frac{\pi^2}{4}-\frac{9\pi^2}{16}=-\frac{-5\pi^2}{16}$
$f(1)=\frac{\pi^2}{4}-\frac{\pi^2}{16}=\frac{3\pi^2}{16}$
But I am not able to show that $f(x)$ is monotonic.
$f'(x)=\frac{2\sin^{-1}x}{\sqrt{1-x^2}}+\frac{2\cot^{-1}x}{1+x^2}$
How to go from here.