Let a vertical tower A B have its end A on the level ground. Let C be the mid-point of A B and P be a point…

Let $\angle APC=\alpha$
Given, $AB=2\left(AC\right)=2\left(BC\right)$

Also $AP=2AB\Rightarrow AP=4\left(AC\right)=4\left(BC\right)$

$\Rightarrow \tan \alpha =\frac{AC}{AP}=\frac{AC}{4\left(AC\right)}=\frac{1}{4}$

$\Rightarrow \tan \left(\alpha +\beta \right)=\frac{AB}{AP}=\frac{AB}{2\left(AB\right)}=\frac{1}{2}$

Let, $\tan \beta =x$

$\frac{\tan \alpha +\tan \beta }{1-\tan \alpha \cdot \tan \beta }=\frac{1}{2}$

$\Rightarrow \frac{\frac{1}{4}+x}{1-\frac{x}{4}}=\frac{1}{2}$

$\Rightarrow \frac{1+4x}{4-x}=\frac{1}{2}$

$\Rightarrow 2+8x-4+x=0$

$\Rightarrow 9x-2=0$

$\Rightarrow x=\frac{2}{9}$