For the following two (02) items that follow : From the top of a building, the angles of depression of the top and bottom of a tower of height \(h\) are \(\alpha\) and \(\beta\) respectively. Let \(H\) be the height of the building and \(D\) be the horizontal distance between the building and the tower.
What is \(H^2 + D^2\) equal to?
\(\dfrac{h^2\cos^2\alpha}{\sin^2(\beta-\alpha)}\)
Let the building have height \(H\) and the tower have height \(h\), with horizontal separation \(D\). From the top of the building, \(\tan\alpha=\dfrac{H-h}{D}\) and \(\tan\beta=\dfrac{H}{D}\), giving \(D(\tan\beta-\tan\alpha)=h\) and \(H=D\tan\beta\). Since \(\tan\beta-\tan\alpha=\dfrac{\sin(\beta-\alpha)}{\cos\alpha\cos\beta}\), we get \(D=\dfrac{h\cos\alpha\cos\beta}{\sin(\beta-\alpha)}\); substituting into \(H^2+D^2=D^2\sec^2\beta\) simplifies to \(\dfrac{h^2\cos^2\alpha}{\sin^2(\beta-\alpha)}\).
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