$$\begin{gathered} {\cos }^{-1}\mathrm{x}=\mathrm{\theta }+{\cos }^{-1}\left(\mathrm{y}/2\right) \\ \Rightarrow \mathrm{x}=\mathrm{cos\theta }.\left(\mathrm{y}/2\right)-\mathrm{sin\theta } \frac{\sqrt{1-{\mathrm{y}}^2}}{4} \\ \Rightarrow {\mathrm{x}}^2-2\cos \mathrm{\theta }\left(\mathrm{y}/2\right)\mathrm{x}+{\cos }^2\mathrm{\theta }. \frac{{\mathrm{y}}^2}{4}={\sin }^2\mathrm{\theta }\left(1-\frac{{\mathrm{y}}^2}{4}\right) \\ \Rightarrow 4{\mathrm{x}}^2-4\mathrm{xycos\theta }+{\mathrm{y}}^2=4{\sin }^2\mathrm{\theta } \end{gathered}$$