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$P Q$ and $R S$ are two vertical towers of the same height where $S$ is on the ground $Q$ is above the ground. The line joining the top $P$ and the foot $S$ of the two towers meets the horizontal line through $Q$ at a point A where the angles of elevation of the tops $P$ and $R$ of the two towers are $α$ and $β$ respectively. If $A S = a,$ the height of the towers is

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$\frac{a \cos (β + α)}{\cos β}$

$\frac{a \sin (β + α)}{\sin β}$

$\frac{a \sin (β + α)}{\cos β}$

$\frac{a \cos (β + α)}{\sin β}$

answer is A.

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Detailed Solution

$h = P Q = R S$

$\begin{array}{l} = R B + B S \\ = A B (\tan α + \tan β) \\ = a \cos α (\tan α + \tan β) \\ = a \frac{\sin (α + β)}{\cos β} \end{array}$

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