If `A+B=135^(@)`, then `(1+cotA)(1+cotB)=?`

Options

A. (a) 1
B. (b) 2
C. (c) 3
D. (d) 4

Answer

To solve the problem, we need to find the value of \((1 + \cot A)(1 + \cot B)\) given that \(A + B = 135^\circ\).

### Step-by-step Solution:

1. **Use the cotangent addition formula**:
We know that:
\[ \cot(A + B) = \frac{\cot A \cot B - 1}{\cot A + \cot B} \]
Since \(A + B = 135^\circ\), we can find \(\cot(135^\circ)\):
\[ \cot(135^\circ) = -1 \]

2. **Set up the equation**:
Plugging this into the cotangent addition formula gives us:
\[ -1 = \frac{\cot A \cot B - 1}{\cot A + \cot B} \]

3. **Cross-multiply to eliminate the fraction**:
This leads to:
\[ -(\cot A + \cot B) = \cot A \cot B - 1 \]
Rearranging this gives:
\[ \cot A \cot B + \cot A + \cot B - 1 = 0 \]

4. **Introduce new variables**:
Let \(x = \cot A\) and \(y = \cot B\). The equation becomes:
\[ xy + x + y - 1 = 0 \]

5. **Rearranging the equation**:
We can rearrange this to:
\[ xy + x + y = 1 \]

6. **Express \((1 + \cot A)(1 + \cot B)\)**:
Now, we can expand \((1 + x)(1 + y)\):
\[ (1 + x)(1 + y) = 1 + x + y + xy \]

7. **Substituting the values**:
From our earlier equation \(xy + x + y = 1\), we substitute:
\[ 1 + x + y + xy = 1 + 1 = 2 \]

8. **Final answer**:
Therefore, we find that:
\[ (1 + \cot A)(1 + \cot B) = 2 \]

### Final Result:
\[ \boxed{2} \]