If $\mathrm{\theta }$ is eliminated from the equations $\mathrm{x}=\mathrm{acos}(\mathrm{\theta }-\mathrm{\alpha })$ and $\mathrm{y}=\mathrm{bcos}(\mathrm{\theta }-\mathrm{\beta })\text{, then }\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{b^2}-\frac{2\mathrm{xy}}{ab}\cos (\mathrm{\alpha }-\mathrm{\beta })$ is equal to

$\begin{array}{l}(\mathrm{\alpha }-\mathrm{\beta })=(\mathrm{\theta }-\mathrm{\beta })-(\mathrm{\theta }-\mathrm{\alpha })\\ \Rightarrow \cos (\mathrm{\alpha }-\mathrm{\beta })=\cos (\mathrm{\theta }-\mathrm{\beta })\cos (\mathrm{\theta }-\mathrm{\alpha })+\sin (\mathrm{\theta }-\mathrm{\beta })\sin (\mathrm{\theta }-\mathrm{\alpha })\end{array}$

$=\frac{\mathrm{y}}{\mathrm{b}}\times \frac{\mathrm{x}}{\mathrm{a}}+\sqrt{1-\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}}\sqrt{1-\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}}$

$\begin{array}{l}\Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{\left[\frac{\mathrm{xy}}{\mathrm{ab}}-\cos (\mathrm{\alpha }-\mathrm{\beta })\right]}^2=\left(1-\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}\right)\left(1-\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}\right)\\ \text{ or }\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\frac{{\mathrm{x}}^2{\mathrm{y}}^2}{{\mathrm{a}}^2{\mathrm{b}}^2}+{\cos }^2(\mathrm{\alpha }-\mathrm{\beta })-\frac{2\mathrm{xy}}{\mathrm{ab}}\cos (\mathrm{\alpha }-\mathrm{\beta })\\ \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=1-\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}-\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{x}}^2{\mathrm{y}}^2}{{\mathrm{a}}^2{\mathrm{b}}^2}\end{array}$

$\text{ or }\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}-\frac{2\mathrm{xy}}{\mathrm{ab}}\cos (\mathrm{\alpha }-\mathrm{\beta })={\sin }^2(\mathrm{\alpha }-\mathrm{\beta })$