If tan (A + B) = `sqrt(3)` and tan (A-B) `=(1)/(sqrt(3)) A gt B`, then the value of A is ..................

Answer

To solve the problem, we start with the given equations:

1. \( \tan(A + B) = \sqrt{3} \)
2. \( \tan(A - B) = \frac{1}{\sqrt{3}} \)

### Step 1: Identify angles from the tangent values
From trigonometric values, we know:
- \( \tan(60^\circ) = \sqrt{3} \)
- \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \)

Thus, we can set up the equations based on these angles:
- From \( \tan(A + B) = \sqrt{3} \), we have:
\[ A + B = 60^\circ \quad \text{(Equation 1)} \]
- From \( \tan(A - B) = \frac{1}{\sqrt{3}} \), we have:
\[ A - B = 30^\circ \quad \text{(Equation 2)} \]

### Step 2: Solve the system of equations
Now, we will add Equation 1 and Equation 2 to eliminate \( B \):
\[ (A + B) + (A - B) = 60^\circ + 30^\circ \]
This simplifies to:
\[ 2A = 90^\circ \]
Now, divide both sides by 2:
\[ A = 45^\circ \]

### Step 3: Find the value of \( B \)
Next, we substitute the value of \( A \) back into Equation 1 to find \( B \):
\[ 45^\circ + B = 60^\circ \]
Subtracting \( 45^\circ \) from both sides gives:
\[ B = 15^\circ \]

### Final Answer
Thus, the value of \( A \) is:
\[ \boxed{45^\circ} \]