If cot A = 2 and cot B = 2 , then what is the value of A +B ?

Options

A. `pi//6`
B. `pi`
C. `pi//2`
D. `pi//4`

Explanation

To find the value of \( A + B \) given that \( \cot A = 2 \) and \( \cot B = 2 \), we can use the cotangent addition formula. The formula for \( \cot(A + B) \) is:

\[ \cot(A + B) = \frac{\cot A \cot B - 1}{\cot A + \cot B} \]

### Step-by-Step Solution:

1. **Identify the values of \( \cot A \) and \( \cot B \)**:
\[ \cot A = 2, \quad \cot B = 2 \]

2. **Substitute the values into the cotangent addition formula**:
\[ \cot(A + B) = \frac{\cot A \cdot \cot B - 1}{\cot A + \cot B} \]
Substituting \( \cot A \) and \( \cot B \):
\[ \cot(A + B) = \frac{2 \cdot 2 - 1}{2 + 2} \]

3. **Calculate the numerator and denominator**:
- Numerator:
\[ 2 \cdot 2 - 1 = 4 - 1 = 3 \]
- Denominator:
\[ 2 + 2 = 4 \]

4. **Combine the results**:
\[ \cot(A + B) = \frac{3}{4} \]

5. **Find \( A + B \)**:
To find \( A + B \), we need to determine the angle whose cotangent is \( \frac{3}{4} \). This can be expressed as:
\[ A + B = \cot^{-1}\left(\frac{3}{4}\right) \]

6. **Convert to radians**:
The exact value of \( A + B \) in radians can be approximated or calculated using a calculator. However, we can also express it in terms of known angles if needed.

### Final Answer:
\[ A + B = \cot^{-1}\left(\frac{3}{4}\right) \]