If `A+B+C=180^(@)`, then
`(tanA+tanB+tanC)/(tanAtanBtanC)=?`

Options

A. 0
B. 2
C. 1
D. `-1`

Answer

To solve the problem, we need to find the value of \((\tan A + \tan B + \tan C) / (\tan A \tan B \tan C)\) given that \(A + B + C = 180^\circ\).

### Step-by-Step Solution:

1. **Understanding the Angles**:
Since \(A + B + C = 180^\circ\), we can interpret \(A\), \(B\), and \(C\) as the angles of a triangle.

2. **Using the Tangent Addition Formula**:
We know that for any triangle, the following identity holds:
\[ \tan A + \tan B + \tan C = \tan A \tan B \tan C \]
This identity is derived from the tangent of the sum of angles.

3. **Substituting the Identity**:
From the identity, we can substitute into our original expression:
\[ \frac{\tan A + \tan B + \tan C}{\tan A \tan B \tan C} = \frac{\tan A \tan B \tan C}{\tan A \tan B \tan C} \]

4. **Simplifying the Expression**:
The expression simplifies to:
\[ \frac{\tan A \tan B \tan C}{\tan A \tan B \tan C} = 1 \]

5. **Conclusion**:
Therefore, the value of \(\frac{\tan A + \tan B + \tan C}{\tan A \tan B \tan C}\) is \(1\).

### Final Answer:
\[ \frac{\tan A + \tan B + \tan C}{\tan A \tan B \tan C} = 1 \]