If angles A and B satisfy `(1+tanA)(1+tanB)=2`, then the value of A +B is

Options

A. `-(11pi)/4`
B. `(15pi)/4`
C. `(23pi)/4`
D. `(-17pi)/4`

Answer

To solve the problem, we start with the given equation:

\[ (1 + \tan A)(1 + \tan B) = 2 \]

### Step 1: Expand the left-hand side
We can expand the left-hand side of the equation:

\[ 1 + \tan A + \tan B + \tan A \tan B = 2 \]

### Step 2: Rearrange the equation
Now, we can rearrange the equation to isolate the terms involving \(\tan A\) and \(\tan B\):

\[ \tan A + \tan B + \tan A \tan B = 2 - 1 \]

This simplifies to:

\[ \tan A + \tan B + \tan A \tan B = 1 \]

### Step 3: Use the tangent addition formula
Recall the tangent addition formula:

\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]

From our rearranged equation, we have:

\[ \tan A + \tan B = 1 - \tan A \tan B \]

Substituting this into the tangent addition formula gives:

\[ \tan(A + B) = \frac{1 - \tan A \tan B}{1 - \tan A \tan B} \]

This simplifies to:

\[ \tan(A + B) = 1 \]

### Step 4: Solve for \(A + B\)
The value of \(A + B\) for which \(\tan(A + B) = 1\) is:

\[ A + B = \frac{\pi}{4} + n\pi \quad (n \in \mathbb{Z}) \]

Since we are looking for the principal value, we can take:

\[ A + B = \frac{\pi}{4} \]

### Final Answer
Thus, the value of \(A + B\) is:

\[ \boxed{\frac{\pi}{4}} \]