If `A+B=45^(@)`, then find the value of `tanA+tanB+tanA tanB`.

Answer

To solve the problem where \( A + B = 45^\circ \) and we need to find the value of \( \tan A + \tan B + \tan A \tan B \), we can follow these steps:

### Step 1: Use the sum of angles formula for tangent
We know that:
\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]
Given that \( A + B = 45^\circ \), we can substitute this into the formula:
\[ \tan(45^\circ) = 1 \]
Thus, we have:
\[ 1 = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]

### Step 2: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 1 - \tan A \tan B = \tan A + \tan B \]

### Step 3: Rearranging the equation
Rearranging the equation, we get:
\[ \tan A + \tan B + \tan A \tan B = 1 \]

### Step 4: Conclusion
Therefore, the value of \( \tan A + \tan B + \tan A \tan B \) is:
\[ \boxed{1} \]

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