The angle of elevation of a tower of heighth from a point A due south of it is x and from a point B due east of A is y. If AB = 2, then which one of the following is correct?

Options

A. `h ^(2) (cot ^(2) y - cot ^(2) x ) = 2 ^(2)`
B. `z ^(2) ( cot ^(2) y - cot ^(2) x ) = h ^(2)`
C. `h ^(2) ( cot ^(2) y - cot ^(2) x ) = h ^(2)`
D. `h ^(2) ( tan ^(2) y - tan ^(2) x ) = h ^(2)`

Answer

To solve the problem, we need to analyze the situation involving the tower, points A and B, and the angles of elevation.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- Let the height of the tower be \( h \).
- Point A is located due south of the tower, and the angle of elevation from A to the top of the tower is \( x \).
- Point B is located due east of point A, and the angle of elevation from B to the top of the tower is \( y \).
- The distance between points A and B is given as \( AB = 2 \).

2. **Using Trigonometric Ratios**:
- From point A, we can use the tangent function to express the height of the tower:
\[ \tan(x) = \frac{h}{d_A} \]
where \( d_A \) is the distance from point A to the base of the tower.

- From point B, we can express the height of the tower similarly:
\[ \tan(y) = \frac{h}{d_B} \]
where \( d_B \) is the distance from point B to the base of the tower.

3. **Relating Distances**:
- Since point B is due east of point A and the distance \( AB = 2 \), we have:
\[ d_B = d_A + 2 \]

4. **Substituting Distances**:
- Substitute \( d_B \) in the second equation:
\[ \tan(y) = \frac{h}{d_A + 2} \]

5. **Equating Heights**:
- From both equations, we can express \( h \) in terms of \( d_A \) and \( d_B \):
\[ h = d_A \tan(x) \]
\[ h = (d_A + 2) \tan(y) \]

6. **Setting the Equations Equal**:
- Since both expressions equal \( h \), we set them equal to each other:
\[ d_A \tan(x) = (d_A + 2) \tan(y) \]

7. **Solving for \( d_A \)**:
- Rearranging gives:
\[ d_A \tan(x) = d_A \tan(y) + 2 \tan(y) \]
\[ d_A (\tan(x) - \tan(y)) = 2 \tan(y) \]
\[ d_A = \frac{2 \tan(y)}{\tan(x) - \tan(y)} \]

8. **Conclusion**:
- The relationship between the angles and the distances can be summarized as:
\[ d_A = \frac{2 \tan(y)}{\tan(x) - \tan(y)} \]
- This indicates that the angles \( x \) and \( y \) are related through the distances and the height of the tower.