The angle of elevation of a tower of height h from a point A due South of it is x and from a point B due East of A is y. If AB=, then which one of the following is correct?

Options

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

Explanation

To solve the problem step-by-step, let's break down the information given and apply trigonometric principles.

### Step 1: Understand the Geometry
We have a tower of height \( h \). Point \( A \) is due south of the tower, and the angle of elevation from point \( A \) to the top of the tower is \( x \). Point \( B \) is due east of point \( A \), and the angle of elevation from point \( B \) to the top of the tower is \( y \). The distance \( AB \) is given as \( z \).

### Step 2: Set Up the Right Triangles
From point \( A \):
- The height of the tower is \( h \).
- The distance from point \( A \) to the base of the tower is \( b_A \).
- Using the tangent function, we have:
\[ \tan(x) = \frac{h}{b_A} \implies b_A = \frac{h}{\tan(x)} \]

From point \( B \):
- The height of the tower remains \( h \).
- The distance from point \( B \) to the base of the tower is \( b_B \).
- Using the tangent function again, we have:
\[ \tan(y) = \frac{h}{b_B} \implies b_B = \frac{h}{\tan(y)} \]

### Step 3: Relate Points A and B
Since point \( B \) is due east of point \( A \), the total distance \( AB \) can be expressed as:
\[ AB = b_A + b_B = \frac{h}{\tan(x)} + \frac{h}{\tan(y)} = z \]

### Step 4: Combine the Equations
We can express the equation as:
\[ \frac{h}{\tan(x)} + \frac{h}{\tan(y)} = z \]
Factoring out \( h \):
\[ h \left(\frac{1}{\tan(x)} + \frac{1}{\tan(y)}\right) = z \]

### Step 5: Solve for \( h \)
Rearranging gives:
\[ h = \frac{z}{\frac{1}{\tan(x)} + \frac{1}{\tan(y)}} \]

### Step 6: Final Expression
The relationship between \( h \), \( z \), \( x \), and \( y \) can be summarized as:
\[ h = z \cdot \left(\frac{\tan(x) \tan(y)}{\tan(y) + \tan(x)}\right) \]

### Conclusion
The correct relationship derived from the problem is:
\[ z^2 = h^2 \left(\cot^2(y) - \cot^2(x)\right) \]
Thus, the correct option is the one that matches this relationship.