Shots are fired simultaneously from the top and bottom of a vertical cliff at angles $\alpha$ and $\beta$ and they strike an object simultaneously at the same point. The horizontal distance from the object from the cliff is ‘a’. 

Coordinates of P w.r.t O and O’ as origins are (a, n) and (a, - m)
The height of the cliff = h = m + n
$\begin{array}{l}y=x\tan \alpha -\frac{g{x}^2}{2u^2{\cos }^2\alpha }\therefore -m=a\tan \alpha -\frac{g{a}^2}{2u^2{\cos }^2\alpha }\\ n=a\tan \beta -\frac{g{a}^2}{2v^2{\cos }^2\beta }\therefore (m+n)=a(\tan \beta -\tan \alpha )\end{array}$

Because  $u=\cos \alpha =v\cos \beta$
 Because both the shots reach P simultaneously. This is possible only if the horizontal  
 components of the velocities are equal
$\therefore h=a(\tan \beta -\tan \alpha )=a(\frac{1}{\cot \beta }-\frac{1}{\cot \alpha })=\frac{a(\cot \alpha -\cot \beta )}{\cot \alpha \cot \beta }$