The height of a tower is h and the angle of elevation of the top of the tower is `alpha ` . On moving a distance h/2 towards the tower, the angle of elevation becomes ` beta` . What is the value of cot ` alpha - cot beta`

Options

A. `(1)/(2)`
B. ` (2)/(3)`
C. 1
D. 2

Answer

To solve the problem, we need to find the value of \( \cot \alpha - \cot \beta \) given the height of the tower \( h \), the angle of elevation \( \alpha \) from a distance \( x \), and the angle of elevation \( \beta \) when the observer moves \( \frac{h}{2} \) closer to the tower.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- Let the height of the tower be \( h \).
- Let the distance from the base of the tower to the observer initially be \( x \).
- After moving \( \frac{h}{2} \) towards the tower, the new distance from the tower becomes \( x - \frac{h}{2} \).

2. **Using Trigonometric Ratios**:
- From the first position (distance \( x \)), we have:
\[ \tan \alpha = \frac{h}{x} \]
Therefore,
\[ \cot \alpha = \frac{x}{h} \]

- From the second position (distance \( x - \frac{h}{2} \)), we have:
\[ \tan \beta = \frac{h}{x - \frac{h}{2}} \]
Therefore,
\[ \cot \beta = \frac{x - \frac{h}{2}}{h} \]

3. **Finding \( \cot \alpha - \cot \beta \)**:
- Now, substituting the values of \( \cot \alpha \) and \( \cot \beta \):
\[ \cot \alpha - \cot \beta = \frac{x}{h} - \frac{x - \frac{h}{2}}{h} \]

4. **Simplifying the Expression**:
- Combine the fractions:
\[ \cot \alpha - \cot \beta = \frac{x - (x - \frac{h}{2})}{h} \]
\[ = \frac{\frac{h}{2}}{h} \]
\[ = \frac{1}{2} \]

5. **Final Result**:
- Therefore, the value of \( \cot \alpha - \cot \beta \) is:
\[ \cot \alpha - \cot \beta = \frac{1}{2} \]