A vertical tower stands on a horizontal plane and is surmounted by a vertical flagstaff of height h. At a point on the plane, the angle of elevation of the bottom of the flagstaff is α and that of the top of the flagstaff is β. The height of the tower is :

1. $\Big(\dfrac{h \tan \beta}{\tan \beta - \tan \alpha}\Big)$

2. $\Big(\dfrac{h \sin \beta}{\tan \beta - \tan \alpha}\Big)$

3. $\Big(\dfrac{h \cot \alpha}{\cot \beta - \cot \alpha}\Big)$

4. $\Big(\dfrac{h \tan \alpha}{\tan \beta - \tan \alpha}\Big)$

Let the height of the tower (BC) be x and the height of the flagstaff (BA) be h.

Let P be the point on ground from foot of tower at distance d.

In triangle PCB,

$\Rightarrow \tan \alpha = \dfrac{x}{d} \\[1em] \Rightarrow d = \dfrac{x}{\tan \alpha} \text{ …..(1)}$

In triangle PCA,

$\Rightarrow \tan \beta = \dfrac{x + h}{d} \\[1em] \Rightarrow d = \dfrac{x + h}{\tan \beta} \text{ ……..(2)}$

From (1) and (2), we get :

$\dfrac{x}{\tan \alpha} = \dfrac{x + h}{\tan \beta}$

x tan β = (x + h)tan α

x tan β = x tan α + h tan α

h tan α = x tan β - x tan α

h tan α = x(tan β - tan α)

x = $\dfrac{h \tan \alpha}{\tan \beta - \tan \alpha}$

Hence, option 4 is the correct option.