A tower subtends angles α , 2 α and 3 α respectively at points, A , B and C (all points lying on the same side on a horizontal line through the foot of the tower), then the value of A B B C is equal to

Step-by-step solution

Let, the height of the tower is P Q = h Where, Q is the foot of the tower and P is the top of the tower, then Q A = h cot ⁡ α Q B = h cot ⁡ 2 α Q C = h cot ⁡ 3 α A B B C = Q A - Q B Q B - Q C = cot ⁡ α - cot ⁡ 2 α cot ⁡ 2 α - cot ⁡ 3 α = cos ⁡ α sin ⁡ α - cos ⁡ 2 α sin ⁡ 2 α cos ⁡ 2 α sin ⁡ 2 α - cos ⁡ 3 α sin ⁡ 3 α = sin ⁡ 2 α cos ⁡ α - cos ⁡ 2 α sin ⁡ α sin ⁡ 3 α cos ⁡ 2 α - cos ⁡ 3 α sin ⁡ 2 α × sin ⁡ 3 α sin ⁡ α = sin ⁡ 3 α sin ⁡ α = 3 - 4 s i n 2 α = 1 + 2 cos ⁡ 2 α