A tower AB leans towards west making an angle `alpha ` with the vertical . The anlgular elevation of B , the topmost point of the tower is `beta` as obsreved from a point C due east of A at distance d from A.If the angular elevation of B from a pont D at a distance 2d due east of C is `gamma` , then prove that 2 tan `alpha` = cot `gamma` -3cot `beta`

Options

A. `2 tan alpha = 2 cot beta - cot gamma `
B. `2 tan alpha = 3 cot beta - cot gamma`
C. ` tan alpha = cot beta - cot gamma`
D. None of these

Explanation

Observe the following figure

Using m-n theorem in triangle ABD we have
`(d+2d ) cot Beta = d cot alpha - 2d cot ( 90+ alpha)`
` rArr 3d cot beta = cot alpha + 2d tan alpha `
`therefore 2 tan alpha = 3 cot beta - cot alpha `