$A$ vertical tower stands on a horizontal plane and is surmounted by a vertical flag staff of height $h$. At a point on the plane,the angles of elevation of the bottom and the top of the flag staff are $\alpha$ and $\beta$,respectively. Prove that the height of the tower is $\left(\frac{h \tan \alpha}{\tan \beta-\tan \alpha}\right)$.

(A) Let the height of the tower be $H$ and the distance from the point to the base of the tower be $x$.
Given that,the height of the flag staff is $h$ and the angles of elevation of the bottom and top of the flag staff are $\alpha$ and $\beta$ respectively.
In $\triangle PRO$,$\tan \alpha = \frac{PO}{RO} = \frac{H}{x} \implies x = \frac{H}{\tan \alpha}$ .....$(i)$
In $\triangle FRO$,$\tan \beta = \frac{FO}{RO} = \frac{FP + PO}{RO} = \frac{h + H}{x} \implies x = \frac{h + H}{\tan \beta}$ .....$(ii)$
Equating $(i)$ and $(ii)$:
$\frac{H}{\tan \alpha} = \frac{h + H}{\tan \beta}$
$H \tan \beta = (h + H) \tan \alpha$
$H \tan \beta = h \tan \alpha + H \tan \alpha$
$H \tan \beta - H \tan \alpha = h \tan \alpha$
$H(\tan \beta - \tan \alpha) = h \tan \alpha$
$H = \frac{h \tan \alpha}{\tan \beta - \tan \alpha}$
Hence,the height of the tower is $\frac{h \tan \alpha}{\tan \beta - \tan \alpha}$.