If `A+B=(pi)/(4),then (tanA+1)(tanB+1)` is equal to

Options

A. 1
B. 2
C. `sqrt3`
D. `-1`

Answer

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

### Step 1: Use the given equation
We start with the equation:
\[ A + B = \frac{\pi}{4} \]

### Step 2: Apply the tangent function
Taking the tangent of both sides, we have:
\[ \tan(A + B) = \tan\left(\frac{\pi}{4}\right) \]
Since \( \tan\left(\frac{\pi}{4}\right) = 1 \), we can write:
\[ \tan(A + B) = 1 \]

### Step 3: Use the tangent addition formula
Using the tangent addition formula, we have:
\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]
Setting this equal to 1 gives us:
\[ \frac{\tan A + \tan B}{1 - \tan A \tan B} = 1 \]

### Step 4: Cross-multiply to eliminate the fraction
Cross-multiplying results in:
\[ \tan A + \tan B = 1 - \tan A \tan B \]

### Step 5: Rearranging the equation
Rearranging gives us:
\[ \tan A + \tan B + \tan A \tan B = 1 \]

### Step 6: Adding 1 to both sides
Now, add 1 to both sides:
\[ \tan A + \tan B + \tan A \tan B + 1 = 2 \]

### Step 7: Factor the left-hand side
The left-hand side can be factored as:
\[ (\tan A + 1)(\tan B + 1) = 2 \]

### Conclusion
Thus, we find that:
\[ (\tan A + 1)(\tan B + 1) = 2 \]

### Final Answer
The value of \( (tan A + 1)(tan B + 1) \) is \( 2 \).

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