If `A + B = 45^(@), " then " (cot A - 1) ( cot B - 1)` is

Options

A. 1
B. `(1)/(2)`
C. `-1`
D. 2

Answer

To solve the problem, we need to find the value of \((\cot A - 1)(\cot B - 1)\) given that \(A + B = 45^\circ\).

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ A + B = 45^\circ \]

2. **Take the cotangent of both sides**:
\[ \cot(A + B) = \cot(45^\circ) \]
Since \(\cot(45^\circ) = 1\), we have:
\[ \cot(A + B) = 1 \]

3. **Use the cotangent addition formula**:
The formula for \(\cot(A + B)\) is:
\[ \cot(A + B) = \frac{\cot A \cot B - 1}{\cot A + \cot B} \]
Setting this equal to 1 gives:
\[ \frac{\cot A \cot B - 1}{\cot A + \cot B} = 1 \]

4. **Cross-multiply to eliminate the fraction**:
\[ \cot A \cot B - 1 = \cot A + \cot B \]

5. **Rearranging the equation**:
\[ \cot A \cot B - \cot A - \cot B + 1 = 0 \]

6. **Factoring the equation**:
We can rewrite it as:
\[ (\cot A - 1)(\cot B - 1) = 2 \]
This is derived from rearranging the terms appropriately.

7. **Conclusion**:
Therefore, we find that:
\[ (\cot A - 1)(\cot B - 1) = 2 \]

### Final Answer:
The value of \((\cot A - 1)(\cot B - 1)\) is \(2\).