From the top of a spire the angle of depression of the top and bottom of a tower of height h are `theta and phi` respectively. Then the height of the spire and its horizontal distance from the tower are respectively `(hcosthetasinphi)/(sin(theta+phi)) and (hcosthetacosphi)/(sin(theta+phi))`

Answer

To solve the problem, we will use the concepts of trigonometry, specifically the tangent of angles and the relationships between angles of elevation and depression.

### Step 1: Understand the Geometry
- Let the height of the tower be \( h \).
- Let the height of the spire be \( H \).
- Let the horizontal distance from the base of the tower to the base of the spire be \( d \).
- The angle of depression to the top of the tower is \( \theta \) and to the bottom of the tower is \( \phi \).

### Step 2: Set Up the Relationships
From the top of the spire:
1. The angle of depression to the top of the tower (\( \theta \)):
\[ \tan(\theta) = \frac{H - h}{d} \]
Rearranging gives:
\[ H - h = d \tan(\theta) \quad \text{(1)} \]

2. The angle of depression to the bottom of the tower (\( \phi \)):
\[ \tan(\phi) = \frac{H}{d} \]
Rearranging gives:
\[ H = d \tan(\phi) \quad \text{(2)} \]

### Step 3: Substitute and Solve for \( H \) and \( d \)
From equation (2), we can express \( d \):
\[ d = \frac{H}{\tan(\phi)} \quad \text{(3)} \]

Now substitute equation (3) into equation (1):
\[ H - h = \left(\frac{H}{\tan(\phi)}\right) \tan(\theta) \]
Multiplying both sides by \( \tan(\phi) \):
\[ (H - h) \tan(\phi) = H \tan(\theta) \]
Expanding gives:
\[ H \tan(\phi) - h \tan(\phi) = H \tan(\theta) \]
Rearranging gives:
\[ H \tan(\phi) - H \tan(\theta) = h \tan(\phi) \]
Factoring out \( H \):
\[ H (\tan(\phi) - \tan(\theta)) = h \tan(\phi) \]
Thus,
\[ H = \frac{h \tan(\phi)}{\tan(\phi) - \tan(\theta)} \quad \text{(4)} \]

### Step 4: Find the Horizontal Distance \( d \)
Now substitute \( H \) from equation (4) back into equation (3):
\[ d = \frac{H}{\tan(\phi)} = \frac{h \tan(\phi)}{\tan(\phi) - \tan(\theta) \tan(\phi)} \]
This simplifies to:
\[ d = \frac{h}{\tan(\phi) - \tan(\theta)} \quad \text{(5)} \]

### Step 5: Final Expressions
Now we can express \( H \) and \( d \) in terms of \( h \), \( \theta \), and \( \phi \):
1. Height of the spire \( H \):
\[ H = \frac{h \tan(\phi)}{\tan(\phi) - \tan(\theta)} \]
2. Horizontal distance \( d \):
\[ d = \frac{h}{\tan(\phi) - \tan(\theta)} \]

### Conclusion
Thus, the height of the spire and its horizontal distance from the tower are given by:
- Height of the spire: \( \frac{h \cos(\theta) \sin(\phi)}{\sin(\theta + \phi)} \)
- Horizontal distance: \( \frac{h \cos(\theta) \cos(\phi)}{\sin(\theta + \phi)} \)