A tower is standing in the centre of an
elliptic field. If Adya observes that the angle of elevation
of the top of the tower at an extremity of the major axis of
the field is $\alpha$, at its focus is $\beta$ and an extremity of the minor
axis is $\gamma$, then 

Let the equation of the elliptic field be

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

$OP$ be the tower of height $h$ at $O$, the centre of the field, Let
$A$ & $B$ be extremities of major and minor axis respectively
and $S$be a focus, then

$\begin{array}{l}OA=h\cot \alpha =a\\ OS=h\cot \beta =ae\\ OB=h\cot \gamma =b\end{array}$

Since      $b^2=a^2\left(1-e^2\right)$

$\Rightarrow {\cot }^2\gamma ={\cot }^2\alpha -{\cot }^2\beta$.