Prove that its inclination is given by cot = b cot

Prove that its inclination is given by cot = b cot - a, Mathematics

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Two stations due south of a leaning tower which leans towards the north are at distances a and b from its foot. If α , β be the elevations of the top of the tower from these stations, prove that its inclination ? is given by cot ? = b cot α - a cot β/b-a .

Ans: Let AE = x, BE = h

Tan φ =BE/AE =h/x

x= h x (1/tan φ)

x = h cot φ ---------------1

tan α =BE/CE = h /a + x

a + x = h cot α

x= h cot α - a --------------2

tan β = BE/DE = h/b + x

b+x = h cot β

x = h cot β - b ---------------3

from 1 and 2

h cot φ = h cot α - a

h ( cot φ + cot α ) = a

h = a /- cot φ + cot α -------------4

from 1 and 3

h cot φ = h cot β - b

h ( cot φ - cot β) = b

h= b/- cot φ + cot β

from 4 and 5

a/- cot φ + cot α = b/- cot φ + cot β

a (cot β - cot φ ) = b ( cot α - cot φ )

- a cot φ + b cot φ = b cot α - a cot β

(b - a) cot φ = b cot α - a cot β

cot φ = b cot α - a cot β/b - a

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