The elevation of a tower due North of a station A is `alpha` and at another station B due West of A it is `beta`. The height of the tower is _____.

Answer

To find the height of the tower given the angles of elevation from two different points, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Points and Angles**:
- Let the height of the tower be \( H \).
- Let \( A \) be the point due south of the tower and \( B \) be the point due east of the tower.
- The angle of elevation from point \( A \) is \( \alpha \) and from point \( B \) is \( \beta \).

2. **Set Up the Right Triangles**:
- From point \( A \), we can form a right triangle \( APQ \) where:
- \( PQ \) is the height of the tower \( H \).
- \( PA \) is the horizontal distance from point \( A \) to the base of the tower.
- From point \( B \), we can form another right triangle \( BPQ \) where:
- \( PB \) is the horizontal distance from point \( B \) to the base of the tower.

3. **Use Trigonometric Ratios**:
- From triangle \( APQ \):
\[ \tan(\alpha) = \frac{H}{PA} \implies PA = \frac{H}{\tan(\alpha)} \]
- From triangle \( BPQ \):
\[ \tan(\beta) = \frac{H}{PB} \implies PB = \frac{H}{\tan(\beta)} \]

4. **Apply Pythagorean Theorem**:
- In triangle \( PAB \), we can apply the Pythagorean theorem:
\[ AB^2 = PA^2 + PB^2 \]
- Substituting \( PA \) and \( PB \):
\[ AB^2 = \left(\frac{H}{\tan(\alpha)}\right)^2 + \left(\frac{H}{\tan(\beta)}\right)^2 \]

5. **Simplify the Equation**:
- This can be rewritten as:
\[ AB^2 = H^2 \left(\frac{1}{\tan^2(\alpha)} + \frac{1}{\tan^2(\beta)}\right) \]
- Rearranging gives:
\[ H^2 = AB^2 \left(\tan^2(\alpha) \tan^2(\beta)\right) \left(\frac{1}{\tan^2(\alpha) + \tan^2(\beta)}\right) \]

6. **Final Expression for Height**:
- Taking the square root gives us the height \( H \):
\[ H = \frac{AB \cdot \tan(\alpha) \cdot \tan(\beta)}{\sqrt{\tan^2(\alpha) + \tan^2(\beta)}} \]

### Final Answer:
The height of the tower is:
\[ H = \frac{AB \cdot \tan(\alpha) \cdot \tan(\beta)}{\sqrt{\tan^2(\alpha) + \tan^2(\beta)}} \]