If (1 + tan A) (1 + tanB) = 2, then { A + B) is equal to

Options

A. `pi/2`
B. `pi/3`
C. `pi/4`
D. `pi/6`

Answer

To solve the equation \( (1 + \tan A)(1 + \tan B) = 2 \) and find the value of \( A + B \), we can follow these steps:

### Step 1: Expand the left-hand side
We start by expanding the left-hand side of the equation:
\[ (1 + \tan A)(1 + \tan B) = 1 + \tan A + \tan B + \tan A \tan B \]
So, we can rewrite the equation as:
\[ 1 + \tan A + \tan B + \tan A \tan B = 2 \]

### Step 2: Simplify the equation
Next, we simplify the equation by subtracting 1 from both sides:
\[ \tan A + \tan B + \tan A \tan B = 1 \]

### Step 3: Use the tangent addition formula
We know that:
\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]
From our equation, we can see that:
\[ \tan A + \tan B = 1 - \tan A \tan B \]
Substituting this into the tangent addition formula gives us:
\[ \tan(A + B) = \frac{1 - \tan A \tan B}{1 - \tan A \tan B} \]
This simplifies to:
\[ \tan(A + B) = 1 \]

### Step 4: Solve for \( A + B \)
The equation \( \tan(A + B) = 1 \) implies that:
\[ A + B = 45^\circ \quad \text{(or any angle of the form \( 45^\circ + n \cdot 180^\circ \))} \]

### Step 5: Convert to radians
Since the question may require the answer in radians, we convert \( 45^\circ \) to radians:
\[ A + B = 45^\circ = \frac{\pi}{4} \text{ radians} \]

### Final Answer
Thus, we conclude that:
\[ A + B = \frac{\pi}{4} \]