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

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​ cos − 1 x − ( 2 π ​ − cos − 1 y ) = α cos − 1 x + cos − 1 y = 2 π ​ + α α ∈ [ − 2 π ​ , π ] , 2 π ​ + α ∈ [ 0 , 2 3 π ​ ] cos − 1 ( x y − 1 − x 2 ​ 1 − y 2 ​ ) = 2 π ​ + α x y − 1 − x 2 ​ 1 − y 2 ​ = − sin α ( x y + sin α ) = ( 1 − x 2 ) ( 1 − y 2 ) x 2 y 2 + 2 x y sin a + sin 2 a = 1 − x 2 − y 2 + x 2 y 2 x 2 + y 2 + 2 x y sin α = 1 − sin 2 α x 2 + y 2 + 2 x y sin α = cos 2 α ​ Min. value of cos 2 α = 0 At α = 2 π ​

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