If cos(A - B) = `(3)/(5)` and tanA tanB = 2, then the value of cosA cosB is _____

Answer

To solve the problem, we need to find the value of \( \cos A \cos B \) given that \( \cos(A - B) = \frac{3}{5} \) and \( \tan A \tan B = 2 \).

### Step-by-step Solution:

1. **Use the Cosine Difference Identity**:
We know that:
\[ \cos(A - B) = \cos A \cos B + \sin A \sin B \]
Given that \( \cos(A - B) = \frac{3}{5} \), we can write:
\[ \cos A \cos B + \sin A \sin B = \frac{3}{5} \tag{1} \]

2. **Express \( \sin A \sin B \) in terms of \( \cos A \cos B \)**:
We are given that \( \tan A \tan B = 2 \). Recall that:
\[ \tan A = \frac{\sin A}{\cos A} \quad \text{and} \quad \tan B = \frac{\sin B}{\cos B} \]
Therefore,
\[ \tan A \tan B = \frac{\sin A \sin B}{\cos A \cos B} = 2 \]
This implies:
\[ \sin A \sin B = 2 \cos A \cos B \tag{2} \]

3. **Substitute Equation (2) into Equation (1)**:
Now we substitute \( \sin A \sin B \) from Equation (2) into Equation (1):
\[ \cos A \cos B + 2 \cos A \cos B = \frac{3}{5} \]
This simplifies to:
\[ 3 \cos A \cos B = \frac{3}{5} \]

4. **Solve for \( \cos A \cos B \)**:
Dividing both sides by 3 gives:
\[ \cos A \cos B = \frac{3}{5} \cdot \frac{1}{3} = \frac{1}{5} \]

Thus, the value of \( \cos A \cos B \) is \( \frac{1}{5} \).

### Final Answer:
\[ \cos A \cos B = \frac{1}{5} \]