If `sinthetaxxcostheta=(1)/(2)`. The value of `sintheta-costheta` is where `0^(@)ltthetalt90^(@)`

Options

A. a.0
B. b.`sqrt(2)`
C. c.2
D. d.1

Answer

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

\[ \sin \theta \cdot \cos \theta = \frac{1}{2} \]

### Step 1: Use the double angle identity
We know that:

\[ \sin(2\theta) = 2 \sin \theta \cos \theta \]

From this, we can rewrite the equation:

\[ \sin(2\theta) = 2 \cdot \frac{1}{2} = 1 \]

### Step 2: Solve for \(2\theta\)
The equation \(\sin(2\theta) = 1\) implies that:

\[ 2\theta = 90^\circ + n \cdot 360^\circ \quad (n \in \mathbb{Z}) \]

Since we are looking for \(0^\circ < \theta < 90^\circ\), we can set \(n = 0\):

\[ 2\theta = 90^\circ \]

### Step 3: Find \(\theta\)
Dividing both sides by 2 gives:

\[ \theta = 45^\circ \]

### Step 4: Calculate \(\sin \theta - \cos \theta\)
Now we need to find the value of \(\sin \theta - \cos \theta\):

\[ \sin(45^\circ) - \cos(45^\circ) \]

Using the known values:

\[ \sin(45^\circ) = \frac{\sqrt{2}}{2} \quad \text{and} \quad \cos(45^\circ) = \frac{\sqrt{2}}{2} \]

Thus:

\[ \sin(45^\circ) - \cos(45^\circ) = \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} = 0 \]

### Final Answer
The value of \(\sin \theta - \cos \theta\) is:

\[ \boxed{0} \]

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