If y = (1 + tan A) (1 - tan B), where `A -B=pi/4` then `(y+ 1)^(y + 1)` is equal to

Options

A. 9
B. 4
C. 27
D. 81

Explanation

To solve the problem step by step, we will start with the given equation and the condition provided.

### Step 1: Understand the given equation
We are given:
\[ y = (1 + \tan A)(1 - \tan B) \]
and the condition:
\[ A - B = \frac{\pi}{4} \]

### Step 2: Apply the tangent difference formula
Using the tangent difference identity, we have:
\[ \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \]
Since \( A - B = \frac{\pi}{4} \), we know:
\[ \tan\left(\frac{\pi}{4}\right) = 1 \]
Thus, we can set up the equation:
\[ \frac{\tan A - \tan B}{1 + \tan A \tan B} = 1 \]

### Step 3: Cross-multiply to simplify
Cross-multiplying gives:
\[ \tan A - \tan B = 1 + \tan A \tan B \]

### Step 4: Rearranging the equation
Rearranging the equation, we get:
\[ \tan A - \tan B - \tan A \tan B = 1 \]
This can be rewritten as:
\[ \tan A - \tan B - \tan A \tan B + 1 = 2 \]

### Step 5: Express in terms of y
Now, we can express \( y \) using the rearranged equation:
\[ y = (1 + \tan A)(1 - \tan B) = 1 + \tan A - \tan B - \tan A \tan B \]
From our previous step, we know:
\[ \tan A - \tan B - \tan A \tan B + 1 = 2 \]
Thus:
\[ y = 2 \]

### Step 6: Calculate \( (y + 1)^{(y + 1)} \)
Now that we have \( y = 2 \):
\[ y + 1 = 2 + 1 = 3 \]
So we need to compute:
\[ (y + 1)^{(y + 1)} = 3^3 = 27 \]

### Final Answer
Thus, the final answer is:
\[ (y + 1)^{(y + 1)} = 27 \]

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