If `tanA=(1)/(sqrt(3))andtanB=sqrt(3)`,then find the value of `cosAcosB-sinAsinB`

Answer

To solve the problem, we need to find the value of \( \cos A \cos B - \sin A \sin B \) given that \( \tan A = \frac{1}{\sqrt{3}} \) and \( \tan B = \sqrt{3} \).

### Step 1: Identify angles A and B
We know that:
- \( \tan 30^\circ = \frac{1}{\sqrt{3}} \), so \( A = 30^\circ \).
- \( \tan 60^\circ = \sqrt{3} \), so \( B = 60^\circ \).

### Step 2: Substitute the angles into the expression
We need to evaluate:
\[ \cos A \cos B - \sin A \sin B \]
Substituting \( A = 30^\circ \) and \( B = 60^\circ \):
\[ \cos(30^\circ) \cos(60^\circ) - \sin(30^\circ) \sin(60^\circ) \]

### Step 3: Find the trigonometric values
We know the following values:
- \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \)
- \( \cos(60^\circ) = \frac{1}{2} \)
- \( \sin(30^\circ) = \frac{1}{2} \)
- \( \sin(60^\circ) = \frac{\sqrt{3}}{2} \)

### Step 4: Substitute the values into the expression
Now substituting these values into the expression:
\[ \frac{\sqrt{3}}{2} \cdot \frac{1}{2} - \frac{1}{2} \cdot \frac{\sqrt{3}}{2} \]

### Step 5: Simplify the expression
Calculating each term:
\[ \frac{\sqrt{3}}{4} - \frac{\sqrt{3}}{4} = 0 \]

### Final Answer
Thus, the value of \( \cos A \cos B - \sin A \sin B \) is:
\[ \boxed{0} \]