If in a triangle ABC, `tan A + tan B + tanC` has the value 6, then the value of cot A cot B cot C is

Options

A. `1//6`
B. 1
C. 6
D. none

Answer

To solve the problem, we need to find the value of \( \cot A \cot B \cot C \) given that \( \tan A + \tan B + \tan C = 6 \).

### Step-by-Step Solution:

1. **Understand the relationship between tangent and cotangent**:
We know that:
\[ \cot A = \frac{1}{\tan A}, \quad \cot B = \frac{1}{\tan B}, \quad \cot C = \frac{1}{\tan C} \]

2. **Use the identity for the sum of tangents in a triangle**:
In a triangle, we have the identity:
\[ \tan A + \tan B + \tan C = \tan A \tan B \tan C \]
Given that \( \tan A + \tan B + \tan C = 6 \), we can set this equal to \( \tan A \tan B \tan C \):
\[ \tan A + \tan B + \tan C = \tan A \tan B \tan C = 6 \]

3. **Let \( \tan A = x \), \( \tan B = y \), \( \tan C = z \)**:
Therefore, we have:
\[ x + y + z = 6 \]
and
\[ xyz = 6 \]

4. **Express cotangent in terms of tangent**:
The product of cotangents can be expressed as:
\[ \cot A \cot B \cot C = \frac{1}{\tan A \tan B \tan C} = \frac{1}{xyz} \]

5. **Substituting the value of \( xyz \)**:
Since we know \( xyz = 6 \), we can substitute this into our expression for \( \cot A \cot B \cot C \):
\[ \cot A \cot B \cot C = \frac{1}{6} \]

6. **Final result**:
Therefore, the value of \( \cot A \cot B \cot C \) is:
\[ \cot A \cot B \cot C = \frac{1}{6} \]

### Conclusion:
The value of \( \cot A \cot B \cot C \) is \( \frac{1}{6} \).