From the top of a hill $h$ metres high the angles of depressions of the top and the bottom of a pillar are $\alpha$ and $\beta$ respectively. The height (in metres) of the pillar is

Let $AB$ be a hill whose height is $h$ metres and $CD$ be a pillar of height $h$ metres.
In $\Delta EDB$,
         $\tan \alpha =\frac{h-h^{\prime }}{ED}$              ......(i)
and in $\Delta ACB$,
         $\tan \beta =\frac{h}{AC}=\frac{h}{ED}$              .......(ii)
Eliminate $ED$ from equations (i) and (ii), we get
         $\tan \alpha =\frac{h-h^{\prime }}{\frac{h}{\tan \beta }}$
$\Rightarrow h\frac{\tan \alpha }{\tan \beta }=h-h^{\prime }$
$\Rightarrow h^{\prime }=\frac{h\tan \beta -h\tan \alpha }{\tan \beta }$