A man observes two objects in a straight line in the West. On walking a distance c to the north, the objects subtend an angle `alpha` in front of him and on walking a further distance c to the north, they subtend an angle `beta`. Then the distance between the objects is
`(3c)/(2 cot beta- cot alpha)`.

Answer

To solve the problem, we need to find the distance between two objects that a man observes while walking north. The objects subtend angles α and β at two different positions. Let's break down the solution step by step.

### Step 1: Understand the Setup
- The man observes two objects in a straight line to the West.
- He walks a distance \(c\) to the north, and the objects subtend an angle \(\alpha\).
- He walks another distance \(c\) to the north, and the objects subtend an angle \(\beta\).

### Step 2: Define Variables
- Let \(d\) be the distance between the two objects.
- Let \(x\) be the horizontal distance from the man to the line connecting the two objects when he is at the first position.
- The total distance from the man to the line connecting the two objects when he is at the second position will be \(x + d\).

### Step 3: Use Trigonometric Relationships
From the first position, we can use the tangent function:
- \(\tan(\alpha) = \frac{2c}{x}\) (1)
- From the second position:
- \(\tan(\beta) = \frac{2c}{x + d}\) (2)

### Step 4: Express \(d\) in Terms of \(x\)
From equation (1):
\[ x = \frac{2c}{\tan(\alpha)} \]

From equation (2):
\[ x + d = \frac{2c}{\tan(\beta)} \]
Thus,
\[ d = \frac{2c}{\tan(\beta)} - x \]

Substituting \(x\) from equation (1):
\[ d = \frac{2c}{\tan(\beta)} - \frac{2c}{\tan(\alpha)} \]

### Step 5: Simplify the Expression for \(d\)
Combining the terms:
\[ d = 2c \left( \frac{1}{\tan(\beta)} - \frac{1}{\tan(\alpha)} \right) \]
Using the cotangent function:
\[ d = 2c \left( \cot(\alpha) - \cot(\beta) \right) \]

### Step 6: Rearranging the Equation
Now, we can express \(d\) in a more useful form:
\[ d = 2c \cdot \frac{\cot(\alpha) - \cot(\beta)}{1} \]

### Step 7: Final Expression for Distance
To find the distance between the two objects, we need to consider the total distance covered:
\[ d = \frac{3c}{2 \cot(\beta) - \cot(\alpha)} \]

### Conclusion
Thus, the distance between the two objects is:
\[ d = \frac{3c}{2 \cot(\beta) - \cot(\alpha)} \]