If $f\left(x\right)={\left({\cos }^{-1}x\right)}^2-{\left({\sin }^{-1}x\right)}^2$ the sum of all the possible integral values of  $\frac{4}{{\pi }^2}f\left(x\right)$is

$\begin{array}{l}f\left(x\right)=\frac{\pi }{2}(\frac{\pi }{2}-2{\sin }^{-1}x)\\ =\frac{{\pi }^2}{4}-\pi {\sin }^{-1}x\\ -\frac{{\pi }^2}{4}\le f\left(x\right)\le \frac{3{\pi }^2}{4}\\ -1\le \frac{4}{{\pi }^2}f\left(x\right)\le 3\end{array}$

$\therefore$The possible integral value are $-1,0,1,2,3$