If ${\cos }^{-1}x+{\cos }^{-1}y+{\cos }^{-1}z+{\cos }^{-1}t=4\pi$, then the value of $x^2+y^2+z^2+t^2$ is:

The range of ${\cos }^{-1}\theta$ is $[0,\pi ]$.The sum of four such terms is $4\pi$.This can only happen if each individual term is equal to its maximum value, $\pi$.
$${\cos }^{-1}x=\pi ⟹x=\cos \pi =-1$$
$${\cos }^{-1}y=\pi ⟹y=-1$$
$${\cos }^{-1}z=\pi ⟹z=-1$$
$${\cos }^{-1}t=\pi ⟹t=-1$$
Therefore:
$$x^2+y^2+z^2+t^2=(-1{)}^2+(-1{)}^2+(-1{)}^2+(-1{)}^2=1+1+1+1=4$$