A lamp post standing at a point$A$ on a
circular path of radius $r$subtends an angle $a$ at some point
$B$ on the path, and $AB$ subtends an angle of 45° at any other
point on the path, then height of the lampost is

Let $AP$ be the lampost of height $h$at a point
$A$ on a circular path of radius $r$and centre $C$. Let $B$be the
point on this path such that$\angle PBA = a$.$\Rightarrow AB = h cot a$
(Fig. 27.27)

Since $AB$ subtends an angle 45° at another point of the
path, it subtends an angle of 90° at the centre $C$ so that

$\mathrm{\angle }BCA={90}^{\circ }$

Also $CA=CB=r\Rightarrow AB=\sqrt{2}r$

and then $h\cot \alpha =\sqrt{2}r$

$\Rightarrow h=\sqrt{2}r\tan \alpha$.