The height of a tower is h and the angle of elevation of the top of the tower is a. On moving a distance h/2 towards, the tower, the angle of elevation becomes 0. The value of cotα - cot β is

Here, $$\angle$$ACB = $$\alpha$$ and $$\angle$$ADB = $$\beta$$

AB = tower = $$h$$ metre

and CD = $$\frac{h}{2}$$ metre

From $$\triangle$$ABC

=> $$tan \alpha = \frac{AB}{BC} = \frac{h}{BC}$$

=> $$BC = h cot \alpha$$ ----------Eqn(1)

From $$\triangle$$ABD

=> $$tan \beta = \frac{AB}{BD} = \frac{h}{BC - CD}$$

=> $$tan \beta = \frac{h}{h cot \alpha - \frac{h}{2}}$$

=> $$h cot \alpha - \frac{h}{2} = h cot \beta$$

=> $$h (cot \alpha - cot \beta) = \frac{h}{2}$$

=> $$cot \alpha - cot \beta = \frac{1}{2}$$