Question : If $\sin A-\cos A=\frac{\sqrt{3}-1}{2}$, then the value of $\sin A\cdot \cos A$ is:
Option 1: $\frac{\sqrt{3}}{2}$
Option 2: $\frac{3}{2}$
Option 3: $\frac{\sqrt{3}}{4}$
Option 4: $\frac{1}{\sqrt{3}}$
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Correct Answer: $\frac{\sqrt{3}}{4}$
Solution :
Given that,
$\sin A-\cos A=\frac{\sqrt3-1}{2}$
Squaring both sides, we get
$\sin^2 A+\cos^2 A-2\sin A \cos A=\frac{1+3-2\sqrt3}{4}$
⇒ $1-2\sin A \cos A=\frac{2-\sqrt3}{2}$
⇒ $2\sin A\cos A=\frac{2-2+\sqrt3}{2}$
⇒ $\sin A \cos A = \frac{\sqrt3}{4}$
Hence, the correct answer is $\frac{\sqrt3}{4}$.
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