The foot of a tower of height h m is in a direct line between two observers A and B. If the angles of elevation of the top of the tower as seen from from A and B are `alpha` and `beta` respectively and if AB=d m, then what is h/d equal to ?

Options

A. `(tan (alpha+beta))/((cot alpha cot beta-1))`
B. `(cot (alpha+beta))/((cot alpha cot beta-1))`
C. `(tan (alpha+beta))/((cot alpha cot beta+1))`
D. `(cot (alpha+beta))/((cot alpha cot beta+1))`

Explanation

To solve the problem step by step, we will use trigonometric relationships involving the angles of elevation from two observers to the top of the tower.

### Step 1: Understand the Setup
We have a tower of height \( h \) meters, and two observers, A and B, who are \( d \) meters apart. The angles of elevation from A and B to the top of the tower are \( \alpha \) and \( \beta \) respectively.

### Step 2: Draw the Diagram
Let’s denote:
- Point D as the foot of the tower.
- Point A as the position of observer A.
- Point B as the position of observer B.
- The distance from A to D as \( x \).
- The distance from B to D as \( d - x \).

### Step 3: Write the Tangent Relationships
From observer A:
\[ \tan(\alpha) = \frac{h}{x} \]
This implies:
\[ x = \frac{h}{\tan(\alpha)} \]

From observer B:
\[ \tan(\beta) = \frac{h}{d - x} \]
This implies:
\[ d - x = \frac{h}{\tan(\beta)} \]

### Step 4: Substitute for \( x \)
Substituting the expression for \( x \) from the first equation into the second equation:
\[ d - \frac{h}{\tan(\alpha)} = \frac{h}{\tan(\beta)} \]

### Step 5: Rearrange the Equation
Rearranging gives:
\[ d = \frac{h}{\tan(\beta)} + \frac{h}{\tan(\alpha)} \]

### Step 6: Factor Out \( h \)
Factoring \( h \) out of the right side:
\[ d = h \left( \frac{1}{\tan(\beta)} + \frac{1}{\tan(\alpha)} \right) \]

### Step 7: Express \( h \) in Terms of \( d \)
Rearranging for \( h \):
\[ h = d \left( \frac{1}{\frac{1}{\tan(\beta)} + \frac{1}{\tan(\alpha)}} \right) \]

### Step 8: Find \( \frac{h}{d} \)
Now, to find \( \frac{h}{d} \):
\[ \frac{h}{d} = \frac{1}{\frac{1}{\tan(\beta)} + \frac{1}{\tan(\alpha)}} \]

### Step 9: Simplify Further
Using the identity for the sum of tangents:
\[ \frac{h}{d} = \frac{\tan(\alpha) \tan(\beta)}{\tan(\alpha) + \tan(\beta)} \]

### Final Result
Thus, we have:
\[ \frac{h}{d} = \frac{\tan(\alpha) \tan(\beta)}{\tan(\alpha) + \tan(\beta)} \]