$$
\begin{aligned}
& \cos ^{-1} x-\cos ^{-1} \frac{y}{2}=\alpha \\
& \cos ^{-1}\left(\frac{x y}{2}+\sqrt{\left(-x^2\right)\left(1-\frac{y^2}{4}\right)}\right)=\alpha \\
& \cos ^{-1}\left(\frac{x y+\sqrt{4-y^2-4 x^2+x^2 y^2}}{2}\right)=\alpha \\
& \Rightarrow 4-y^2-4 x^2+x^2 y^2=4 \cos ^2 \alpha+x^2 y^2-4 x y \cos \alpha \\
& \Rightarrow 4 x^2+y^2-4 x y \cos \alpha=4 \sin ^2 \alpha
\end{aligned}
$$