What is the sign of `cos( x/2) – sin(x/2)` when (i)` 0lt x lt pi/4` (ii) `(pi)/(2)lt xlt pi`

Answer

To determine the sign of the expression \( \cos\left(\frac{x}{2}\right) - \sin\left(\frac{x}{2}\right) \) for the given intervals of \( x \), we will analyze the behavior of the cosine and sine functions in those ranges.

### Step-by-Step Solution:

#### Part (i): When \( 0 < x < \frac{\pi}{4} \)

1. **Determine the range of \( \frac{x}{2} \)**:
- Since \( x \) is between \( 0 \) and \( \frac{\pi}{4} \), we divide by 2 to find the range for \( \frac{x}{2} \):
\[ 0 < \frac{x}{2} < \frac{\pi}{8} \]

2. **Analyze the values of \( \cos\left(\frac{x}{2}\right) \) and \( \sin\left(\frac{x}{2}\right) \)**:
- In the interval \( 0 < \frac{x}{2} < \frac{\pi}{8} \), we know:
- \( \cos\left(\frac{x}{2}\right) \) is decreasing and positive.
- \( \sin\left(\frac{x}{2}\right) \) is increasing and also positive.

3. **Evaluate at \( \frac{x}{2} = \frac{\pi}{8} \)**:
- At \( \frac{\pi}{8} \), both functions are equal to \( \sin\left(\frac{\pi}{8}\right) \) and \( \cos\left(\frac{\pi}{8}\right) \) respectively, but since \( \frac{x}{2} < \frac{\pi}{8} \), we have:
\[ \cos\left(\frac{x}{2}\right) > \sin\left(\frac{x}{2}\right) \]

4. **Conclusion for Part (i)**:
- Therefore, \( \cos\left(\frac{x}{2}\right) - \sin\left(\frac{x}{2}\right) > 0 \) in this interval.
- The sign is **positive**.

#### Part (ii): When \( \frac{\pi}{2} < x < \pi \)

1. **Determine the range of \( \frac{x}{2} \)**:
- Since \( x \) is between \( \frac{\pi}{2} \) and \( \pi \), we divide by 2:
\[ \frac{\pi}{4} < \frac{x}{2} < \frac{\pi}{2} \]

2. **Analyze the values of \( \cos\left(\frac{x}{2}\right) \) and \( \sin\left(\frac{x}{2}\right) \)**:
- In the interval \( \frac{\pi}{4} < \frac{x}{2} < \frac{\pi}{2} \):
- \( \cos\left(\frac{x}{2}\right) \) is decreasing and becomes less than \( \sin\left(\frac{x}{2}\right) \).

3. **Evaluate at \( \frac{x}{2} = \frac{\pi}{4} \)**:
- At \( \frac{\pi}{4} \), both functions are equal, but since \( \frac{x}{2} > \frac{\pi}{4} \), we have:
\[ \cos\left(\frac{x}{2}\right) < \sin\left(\frac{x}{2}\right) \]

4. **Conclusion for Part (ii)**:
- Therefore, \( \cos\left(\frac{x}{2}\right) - \sin\left(\frac{x}{2}\right) < 0 \) in this interval.
- The sign is **negative**.

### Final Answer:
- For \( 0 < x < \frac{\pi}{4} \): The sign is **positive**.
- For \( \frac{\pi}{2} < x < \pi \): The sign is **negative**.