If the line joining the points $\mathrm{A}\left(\mathrm{\alpha }\right)$ and $\mathrm{B}\left(\mathrm{\beta }\right)$ on the ellipse $\frac{{\mathrm{x}}^2}{25}+\frac{{\mathrm{y}}^2}{9}=1$ is a focal chord then one possible value of $\cot \frac{\mathrm{\alpha }}{2}\cot \frac{\mathrm{\beta }}{2}$ is

Given ellipse $\frac{{\mathrm{x}}^2}{25}+\frac{{\mathrm{y}}^2}{9}=1$

Equation of chord joining the points $\mathrm{A}\left(\mathrm{\alpha }\right)$ and $\mathrm{B}\left(\mathrm{\beta }\right)$ is $\frac{\mathrm{x}}{5}\cos \left(\frac{\mathrm{\alpha }+\mathrm{\beta }}{2}\right)+\frac{\mathrm{y}}{3}\sin \left(\frac{\mathrm{\alpha }+\mathrm{\beta }}{2}\right)=\cos \left(\frac{\mathrm{\alpha }-\mathrm{\beta }}{2}\right)$

since it is a focal chord then it passes through the focus (4,0)

$\Rightarrow \frac{4}{5}\cos \left(\frac{\mathrm{\alpha }+\mathrm{\beta }}{2}\right)=\cos \left(\frac{\mathrm{\alpha }-\mathrm{\beta }}{2}\right)$

$\Rightarrow \frac{\cos \left(\frac{\mathrm{\alpha }+\mathrm{\beta }}{2}\right)}{\cos \left(\frac{\mathrm{\alpha }-\mathrm{\beta }}{2}\right)}=\frac{5}{4}$

Now $\frac{\cos \left(\frac{\mathrm{\alpha }+\mathrm{\beta }}{2}\right)+\cos \left(\frac{\mathrm{\alpha }-\mathrm{\beta }}{2}\right)}{\cos \left(\frac{\mathrm{\alpha }+\mathrm{\beta }}{2}\right)-\left(\frac{\mathrm{\alpha }-\mathrm{\beta }}{2}\right)}=\frac{5+4}{5-4}$

$$\begin{gathered} \Rightarrow \frac{2\cos \left(\frac{\mathrm{\alpha }}{2}\right)\cos \left(\frac{\mathrm{\beta }}{2}\right)}{-2\sin \left(\frac{\mathrm{\alpha }}{2}\right)\sin \left(\frac{\mathrm{\beta }}{2}\right)}=9 \\ \Rightarrow \cot \frac{\mathrm{\alpha }}{2} \cot \frac{\mathrm{\beta }}{2}=-9 \end{gathered}$$