$\begin{aligned} & \cos ^{-1} x-\frac{\pi}{2}+\cos ^{-1} y=\alpha \\ & \cos ^{-1} x+\cos ^{-1} y=\frac{\pi}{2}+\alpha \\ & \because \quad \alpha \in\left[\frac{-\pi}{2}, \pi\right] \\ & \text { then } \frac{\pi}{2}+\alpha \in\left(0, \frac{3 \pi}{2}\right) \\ & \cos ^{-1}\left(x y-\sqrt{1-x^2} \sqrt{1-y^2}\right)=\frac{\pi}{2}+\alpha \\ & x y-\sqrt{1-x^2} \sqrt{1-y^2}=-\sin \alpha \\ & x y+\sin \alpha=\sqrt{1-x^2} \sqrt{1-y^2} \\ & x^2 y^2+\sin ^2 \alpha+2 x y \sin \alpha=1-x^2-y^2+x^2 y^2 \\ & \underbrace{x^2+y^2+2 x y \sin \alpha}_E=\cos ^2 \alpha \end{aligned}$ Now, minimum value of $E$ is 0.