If `cos(A-B)=3/5` and `tanAtanB=2` then

Options

A. `cosA cos B = 1/5`
B. `sin A sin B=-2/5`
C. `cosA cos B=-1/5`
D. `sin A sin B= 1//5`

Explanation

To solve the problem, we will follow a systematic approach using trigonometric identities and the information provided.

### Step 1: Write down the given information
We are given:
1. \( \cos(A - B) = \frac{3}{5} \)
2. \( \tan A \tan B = 2 \)

### Step 2: Apply the cosine difference formula
Using the cosine difference formula:
\[ \cos(A - B) = \cos A \cos B + \sin A \sin B \]
Substituting the given value:
\[ \cos A \cos B + \sin A \sin B = \frac{3}{5} \tag{1} \]

### Step 3: Express \( \tan A \tan B \) in terms of sine and cosine
We know that:
\[ \tan A = \frac{\sin A}{\cos A} \quad \text{and} \quad \tan B = \frac{\sin B}{\cos B} \]
Thus:
\[ \tan A \tan B = \frac{\sin A \sin B}{\cos A \cos B} = 2 \]
Rearranging gives:
\[ \sin A \sin B = 2 \cos A \cos B \tag{2} \]

### Step 4: Substitute equation (2) into equation (1)
Now, 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:
\[ 3 \cos A \cos B = \frac{3}{5} \]

### Step 5: Solve for \( \cos A \cos B \)
Dividing both sides by 3:
\[ \cos A \cos B = \frac{1}{5} \tag{3} \]

### Step 6: Find \( \sin A \sin B \)
Using equation (2) again:
\[ \sin A \sin B = 2 \cos A \cos B \]
Substituting the value from equation (3):
\[ \sin A \sin B = 2 \times \frac{1}{5} = \frac{2}{5} \tag{4} \]

### Step 7: Conclusion
From equations (3) and (4), we have:
- \( \cos A \cos B = \frac{1}{5} \)
- \( \sin A \sin B = \frac{2}{5} \)

Thus, the values of \( \cos A \cos B \) and \( \sin A \sin B \) are determined.