Two towers of equal height, h, are x meters apart. From a point between the two towers, the angles of elevation are `alpha` degrees and `beta` degrees. Prove that `h= (x tan alpha tan beta) / (tan alpha +tan beta)` .

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Two towers A and B of equal height, h, are x meters apart. From a point between the two towers, the angles of elevation are `alpha ` degrees and `beta` degrees.

Let the distance of the point from which measurements are being made lie at a distance D from A, the distance of the point from B is x - D).

`tan alpha = h/D` , `tan beta = h/(x - D)`

`(x*tan alpha*tan beta)/(tan alpha +tan beta)`

= `(x*(h/D)*(h/(x - D)))/(h/D + h/(x - D))`

= `h*(x*(1/D)*(1/(x - D)))/(1/D + 1/(x - D))`

= `h*(x*(1/D)*(1/(x - D)))/((x - D + D)/(D*(x - D)))`

= `h*(x*(1/D)*(1/(x - D)))/(x/(D*(x - D)))`

= h

This proves that `h = (x*tan alpha*tan beta)/(tan alpha +tan beta)`

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