A ladder leaning against a vertical wall is inclined at an angle $\alpha$ to the horizontal. The top of the ladder touches the parapet. On moving its foot $a$ feet away from the wall, the ladder now stands inclined at an angle $\beta$ to the horizon, and its top now touching a window. Then the distance of the parapet from the window is $a \cot [(\alpha+\beta)/2]$.

Let the length of the ladder be L, making the initial height of the wall reached equal to L sin alpha and the distance from the wall equal to L cos alpha. In the second position, the height is L sin beta and the distance is L cos beta, meaning the distance the foot moves is a equals L cos beta minus L cos alpha. Using trigonometric sum-to-product formulas to express the vertical distance between the parapet and the window, we get L sin alpha minus L sin beta equals 2L cos((alpha + beta) / 2) sin((alpha minus beta) / 2); dividing this height difference by the horizontal distance a yields the ratio a cot((alpha + beta) / 2), proving the statement is true.

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