From the top of a spire the angle of depression of the top and bottom of a tower of height h are $\theta$ and $\phi$ respectively. Then the height of the spire and its. horizontal distance from the tower are $\dfrac{h\, cos\,\theta\, sin\,\phi}{sin (\theta + \phi)}$ and $\dfrac{h\, cos\,\theta\, cos\,\phi}{sin (\theta + \phi)}$ respectively

Let the total height of the spire be H and the horizontal distance from the tower be x. From the angle of depression phi to the bottom of the tower, we get tan phi equals H divided by x, so H equals x tan phi. From the angle of depression theta to the top of the tower, we get tan theta equals the difference in their heights divided by x, which is H minus h divided by x; substituting H gives x tan theta equals x tan phi minus h. Solving for x yields x equals h divided by tan phi minus tan theta, which simplifies using trigonometric identities to h multiplied by cosine theta and cosine phi divided by sine of phi minus theta. Because the given formula uses sine of the sum of theta and phi instead of the difference, the height and distance expressions are incorrect, making the statement false.

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