If $\mathrm{\theta }$ is eliminated from the equations

$\mathrm{x}=\mathrm{acos}\left(\mathrm{\theta }-\mathrm{\alpha }\right)$ and $\mathrm{y}=\mathrm{acos}\left(\mathrm{\theta }-\mathrm{\beta }\right),$ then

$\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}-\frac{2\mathrm{xy}}{\mathrm{ab}}\cos (\mathrm{\alpha }-\mathrm{\beta })$  is equal to

$\begin{array}{rcl}\mathrm{x}& =& \mathrm{a} \cos (\mathrm{\theta }-\mathrm{\alpha }), \mathrm{y}=\mathrm{b} \cos (\mathrm{\theta }-\mathrm{\beta })\\ & & \frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}-2\frac{\mathrm{xy}}{\mathrm{ab}} \cos (\mathrm{\alpha }-\mathrm{\beta })\\ & =& {\cos }^2(\mathrm{\theta }-\mathrm{\alpha })+{\cos }^2(\mathrm{\theta }-\mathrm{\beta })-2\cos (\mathrm{\theta }-\mathrm{\alpha }) \cos (\mathrm{\theta }-\mathrm{\beta }) \cos (\mathrm{\alpha }-\mathrm{\beta })\\ & =& 1+{\cos }^2(\mathrm{\theta }-\mathrm{\alpha })-{\sin }^2(\mathrm{\theta }-\mathrm{\beta })-\left[\cos (\mathrm{\theta }-(\mathrm{\alpha }+\mathrm{\beta }\left)\right)+\cos (\mathrm{\alpha }-\mathrm{\beta })\right]\cos (\mathrm{\alpha }-\mathrm{\beta })\\ & =& 1+\cos (2\mathrm{\theta }-(\mathrm{\alpha }+\mathrm{\beta }\left)\right).\cos (\mathrm{\alpha }-\mathrm{\beta })-\cos (2\mathrm{\theta }-(\mathrm{\alpha }+\mathrm{\beta }\left)\right) \cos (\mathrm{\alpha }-\mathrm{\beta })-{\cos }^2(\mathrm{\alpha }-\mathrm{\beta })\\ & =& 1-{\cos }^2(\mathrm{\alpha }-\mathrm{\beta })\\ & =& {\sin }^2(\mathrm{\alpha }-\mathrm{\beta })\end{array}$