The top of a pole, placed against a wall
at an angle $α$ with the horizon, just touches the coping, and
when its foot is moved $a m,$ away from the wall and its angle
of inclination is $b$ , it rests on the sill of a window; the vertical
distance of the sill from the coping is

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$a \sin ((α + β) / 2)$

$a \cos ((α + β) / 2)$

$a \cot ((α + β) / 2)$

$a \tan ((α + β) / 2)$

answer is C.

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Detailed Solution

Let $L$ be the coping and $M$ the sill of the window
and $A$ and $B$ be the two positions of the pole such that

$∠ L A X = α$ and $∠ M B X = β \cdot L M = x, X$ being the foot of the
vertical wall. (Fig. 27.45)

Let $A L = B M = l$ and $A B = a$ (given )

then $x = L X - M X = l (\sin α - \sin β)$

$\begin{array}{l} a = B A = B X - A X = l (\cos β - \cos α) . \\ \Rightarrow \frac{x}{a} = \frac{\sin α - \sin β}{\cos β - \cos α} = \frac{2 \cos \frac{α + β}{2} \sin \frac{α - β}{2}}{2 \sin \frac{α + β}{2} \sin \frac{α - β}{2}} \\ \Rightarrow x = a \cot \frac{α + β}{2} . \end{array}$