A chimney leans towards North. At equal distances due north and south of it in a horizontal plane the elevation of the top are `alpha,beta`. The inclination of the chimney to the vertical is
`"tan"^(-1)(sin(alpha-beta))/(2sin alpha sin beta)`
or `"tan"^(-1)(1)/(2)(cot beta-cot alpha)`

Answer

To solve the problem, we need to find the inclination of a chimney leaning towards the North, given the angles of elevation from two points at equal distances to the North and South of the chimney.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- Let the chimney be represented as a line segment PQ, where P is the base and Q is the top of the chimney.
- The angles of elevation from points A (to the North) and B (to the South) are given as α and β, respectively.

2. **Drawing the Diagram**:
- Draw a horizontal line representing the ground level.
- Mark the chimney leaning towards the North.
- From point A, draw a line to point Q making an angle α with the horizontal.
- From point B, draw a line to point Q making an angle β with the horizontal.

3. **Identifying the Angles**:
- At point A, the angle of elevation is α.
- At point B, the angle of elevation is β.
- Let the distance from the chimney to points A and B be equal, denoted as 'd'.

4. **Using Trigonometric Relationships**:
- From point A, the height of the chimney can be expressed as:
\[ h = d \cdot \tan(\alpha) \]
- From point B, the height of the chimney can be expressed as:
\[ h = d \cdot \tan(\beta) \]

5. **Setting the Heights Equal**:
- Since both expressions equal the height of the chimney:
\[ d \cdot \tan(\alpha) = d \cdot \tan(\beta) \]
- This implies:
\[ \tan(\alpha) \neq \tan(\beta) \quad \text{(since d is not zero)} \]

6. **Finding the Inclination**:
- The inclination of the chimney to the vertical can be derived using the formula:
\[ \theta = \tan^{-1}\left(\frac{\sin(\alpha - \beta)}{2 \sin(\alpha) \sin(\beta)}\right) \]
- Alternatively, it can also be expressed as:
\[ \theta = \tan^{-1}\left(\frac{1}{2}(\cot(\beta) - \cot(\alpha))\right) \]

7. **Final Expression**:
- Thus, the inclination of the chimney to the vertical is given by either of the two expressions derived above.