From a window (h metres high above the ground) of a house in a street, the angles of elevation and depression of the top and foot of another house on the opposite side of the street are $\alpha$ and $\beta$ respectively. Is this possible where the height of the opposite house is : $h(1+\tan\,\alpha\,\cot \beta )$ metres.

Let the opposite house have height H and the street width be d, making the vertical distance from the window to the foot of the opposite house h = d tan(beta). The vertical distance from the window to the top of the opposite house is H - h = d tan(alpha). Adding these two equations gives H = d tan(alpha) + d tan(beta). Substituting d = h cot(beta) results in H = h tan(alpha) cot(beta) + h, which factors to h(1 + tan(alpha) cot(beta)). This statement is True.

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