Inverse Trigonometric Functions

Given that the inverse trigonometric function assumes principal values only. Let \(x, y\) be any two real numbers in \([-1,1]\) such that \(\cos ^{-1} x-\sin ^{-1} y=\alpha, \frac{-\pi}{2} \leq \alpha \leq \pi\).
Then, the minimum value of \(x^2+y^2+2 x y \sin \alpha\) is