If `A-B=(pi)/(4)`, then `(1+tanA)(1-tanB)=?`

Options

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

Answer

To solve the problem where \( A - B = \frac{\pi}{4} \) and we need to find \( (1 + \tan A)(1 - \tan B) \), we can follow these steps:

### Step 1: Expand the expression
We start with the expression:
\[ (1 + \tan A)(1 - \tan B) \]
Using the distributive property (also known as the FOIL method for binomials), we can expand this:
\[ = 1 - \tan B + \tan A - \tan A \tan B \]
So, we have:
\[ = 1 + \tan A - \tan B - \tan A \tan B \]

### Step 2: Use the given information
We know that:
\[ A - B = \frac{\pi}{4} \]
Taking the tangent of both sides, we apply the tangent subtraction formula:
\[ \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \]
Since \( A - B = \frac{\pi}{4} \), we have:
\[ \tan\left(\frac{\pi}{4}\right) = 1 \]
Thus, we can set up the equation:
\[ 1 = \frac{\tan A - \tan B}{1 + \tan A \tan B} \]

### Step 3: Cross-multiply
Cross-multiplying gives us:
\[ 1 + \tan A \tan B = \tan A - \tan B \]
Rearranging this, we find:
\[ \tan A - \tan B - \tan A \tan B = 1 \]

### Step 4: Substitute back into the expression
Now, substituting \( \tan A - \tan B - \tan A \tan B \) back into our expanded expression:
\[ 1 + \tan A - \tan B - \tan A \tan B = 1 + 1 = 2 \]

### Final Answer
Thus, the value of \( (1 + \tan A)(1 - \tan B) \) is:
\[ \boxed{2} \]