If `A+C=B,then tanA tanBtanC=`

Options

A. `tanA tanB+tanC`
B. `tanB-tanC-tanA`
C. `tanA+tanC-tanB`
D. `-(tanA tanB+tanC)`

Answer

To solve the problem where \( A + C = B \) and we need to find the value of \( \tan A \tan B \tan C \), we can follow these steps:

### Step-by-Step Solution:

1. **Given Condition**: We start with the equation given in the problem:
\[ A + C = B \]

2. **Apply Tangent**: We apply the tangent function to both sides:
\[ \tan(A + C) = \tan B \]

3. **Use the Tangent Addition Formula**: The formula for the tangent of the sum of two angles is:
\[ \tan(A + C) = \frac{\tan A + \tan C}{1 - \tan A \tan C} \]
Therefore, we can write:
\[ \frac{\tan A + \tan C}{1 - \tan A \tan C} = \tan B \]

4. **Cross Multiply**: We cross-multiply to eliminate the fraction:
\[ \tan A + \tan C = \tan B (1 - \tan A \tan C) \]

5. **Expand the Right Side**: Expanding the right side gives us:
\[ \tan A + \tan C = \tan B - \tan B \tan A \tan C \]

6. **Rearrange the Equation**: We can rearrange this equation to isolate the product \( \tan A \tan B \tan C \):
\[ \tan A + \tan C + \tan B \tan A \tan C = \tan B \]

7. **Isolate the Product**: Now, we can isolate \( \tan A \tan B \tan C \):
\[ \tan A \tan B \tan C = \tan B - \tan A - \tan C \]

8. **Final Result**: Thus, we have:
\[ \tan A \tan B \tan C = \tan B - \tan A - \tan C \]

### Conclusion:
The final expression for \( \tan A \tan B \tan C \) is:
\[ \tan A \tan B \tan C = \tan B - \tan A - \tan C \]