As seen from the top of fort of height `a` metre the angle of depression of the upper and the lower end of a lamp post are `alpha` and `beta`, respectively. The height of the lamp post is :

Options

A. `(a sin (alpha + beta))/(sin alpha . cos beta)`
B. `(a cos (beta + alpha))/(cos alpha . sin beta)`
C. `(a sin (beta -alpha ))/(cos alpha . sin beta)`
D. `(a sin (beta - alpha))/(sin alpha . cos beta)`

Answer

To find the height of the lamp post, we will use the properties of triangles and the angles of depression. Let's denote the following:

- Height of the fort = \( a \) meters
- Height of the lamp post = \( h \) meters
- Angle of depression to the upper end of the lamp post = \( \alpha \)
- Angle of depression to the lower end of the lamp post = \( \beta \)

### Step-by-Step Solution:

1. **Draw the Diagram:**
- Draw a vertical line representing the fort's height \( AD = a \).
- Draw a lamp post \( BC \) such that \( C \) is the top and \( B \) is the bottom of the lamp post.
- The horizontal line from point \( A \) (top of the fort) to point \( D \) (the ground) represents the line of sight.

2. **Identify the Triangles:**
- Triangle \( ABC \) (formed by the top of the fort, the top of the lamp post, and the point directly below the fort).
- Triangle \( ADE \) (formed by the top of the fort, the bottom of the lamp post, and the point directly below the fort).

3. **Using the Angle of Depression:**
- The angle of depression from \( A \) to \( C \) is \( \alpha \) and from \( A \) to \( B \) is \( \beta \).
- By the properties of alternate angles, we have:
- \( \angle ACB = \alpha \)
- \( \angle ADB = \beta \)

4. **Set Up the Equations:**
- For triangle \( ABC \):
\[ \tan(\alpha) = \frac{AD - BC}{BD} = \frac{a - h}{x} \]
Rearranging gives:
\[ x = \frac{a - h}{\tan(\alpha)} \quad \text{(Equation 1)} \]

- For triangle \( ADE \):
\[ \tan(\beta) = \frac{AD}{DE} = \frac{a}{x} \]
Rearranging gives:
\[ x = \frac{a}{\tan(\beta)} \quad \text{(Equation 2)} \]

5. **Equate the Two Expressions for \( x \):**
- From Equation 1 and Equation 2:
\[ \frac{a - h}{\tan(\alpha)} = \frac{a}{\tan(\beta)} \]

6. **Cross Multiply and Solve for \( h \):**
- Cross multiplying gives:
\[ (a - h) \tan(\beta) = a \tan(\alpha) \]
- Expanding and rearranging:
\[ a \tan(\beta) - h \tan(\beta) = a \tan(\alpha) \]
\[ h \tan(\beta) = a \tan(\beta) - a \tan(\alpha) \]
\[ h = a \frac{\tan(\beta) - \tan(\alpha)}{\tan(\beta)} \]

7. **Final Expression for Height \( h \):**
- We can express it in terms of sine and cosine:
\[ h = a \frac{\sin(\beta) \cos(\alpha) - \sin(\alpha) \cos(\beta)}{\cos(\alpha) \sin(\beta)} \]
- Using the sine subtraction formula:
\[ h = a \frac{\sin(\beta - \alpha)}{\cos(\alpha) \sin(\beta)} \]

### Final Answer:
The height of the lamp post is given by:
\[ h = a \frac{\sin(\beta - \alpha)}{\cos(\alpha) \sin(\beta)} \]