If `sinA-cosA=(sqrt(3)-1)/(2)` , then the value of sin A . cos A is

Options

A. `(sqrt(3))/(2)`
B. `(3)/(2)`
C. `(sqrt(3))/(2)`
D. `(1)/(sqrt(3))`

Answer

To solve the problem where \( \sin A - \cos A = \frac{\sqrt{3} - 1}{2} \) and we need to find the value of \( \sin A \cdot \cos A \), we can follow these steps:

### Step 1: Square both sides of the equation
We start with the equation:
\[ \sin A - \cos A = \frac{\sqrt{3} - 1}{2} \]
Now, squaring both sides gives us:
\[ (\sin A - \cos A)^2 = \left(\frac{\sqrt{3} - 1}{2}\right)^2 \]

### Step 2: Expand the left side
Using the identity \( (x - y)^2 = x^2 - 2xy + y^2 \), we expand the left side:
\[ \sin^2 A - 2\sin A \cos A + \cos^2 A \]
Since \( \sin^2 A + \cos^2 A = 1 \), we can substitute this into our equation:
\[ 1 - 2\sin A \cos A = \left(\frac{\sqrt{3} - 1}{2}\right)^2 \]

### Step 3: Calculate the right side
Now we compute the right side:
\[ \left(\frac{\sqrt{3} - 1}{2}\right)^2 = \frac{(\sqrt{3} - 1)^2}{4} = \frac{3 - 2\sqrt{3} + 1}{4} = \frac{4 - 2\sqrt{3}}{4} = 1 - \frac{\sqrt{3}}{2} \]

### Step 4: Set up the equation
Now we have:
\[ 1 - 2\sin A \cos A = 1 - \frac{\sqrt{3}}{2} \]
Subtracting 1 from both sides gives:
\[ -2\sin A \cos A = -\frac{\sqrt{3}}{2} \]

### Step 5: Solve for \( \sin A \cdot \cos A \)
Dividing both sides by -2:
\[ \sin A \cos A = \frac{\sqrt{3}}{4} \]

### Final Answer
Thus, the value of \( \sin A \cdot \cos A \) is:
\[ \boxed{\frac{\sqrt{3}}{4}} \]