If P`(sin theta, 1//sqrt(2)) and Q(1//sqrt(2), cos theta), -pi le theta le pi` are two points on the same side of the line x-y=0, then `theta` belongs to the interval

Options

A. `(-pi//4,pi//4)uu(pi//4,3pi//4)`
B. `(-pi//4,pi//4)`
C. `(pi//4,pi//4)`
D. none of these

Answer

To solve the problem, we need to determine the interval for the angle \( \theta \) such that the points \( P(\sin \theta, \frac{1}{\sqrt{2}}) \) and \( Q(\frac{1}{\sqrt{2}}, \cos \theta) \) lie on the same side of the line \( x - y = 0 \). This line can be rewritten as \( y = x \), which means we need to analyze the conditions under which both points are either above or below this line.

### Step-by-Step Solution:

1. **Identify the Points**:
- Point \( P \) is given as \( P(\sin \theta, \frac{1}{\sqrt{2}}) \).
- Point \( Q \) is given as \( Q(\frac{1}{\sqrt{2}}, \cos \theta) \).

2. **Condition for Same Side**:
- For both points to be on the same side of the line \( x - y = 0 \), we need to check the conditions:
- Both points must satisfy \( x > y \) or both must satisfy \( x < y \).

3. **Case 1: Both Points Above the Line**:
- For point \( P \):
\[ \sin \theta > \frac{1}{\sqrt{2}} \]
- For point \( Q \):
\[ \frac{1}{\sqrt{2}} < \cos \theta \]

4. **Analyzing the Inequalities**:
- The inequality \( \sin \theta > \frac{1}{\sqrt{2}} \) implies:
\[ \theta \in \left( \frac{\pi}{4}, \frac{3\pi}{4} \right) \]
- The inequality \( \cos \theta > \frac{1}{\sqrt{2}} \) implies:
\[ \theta \in \left( -\frac{\pi}{4}, \frac{\pi}{4} \right) \]

5. **Finding the Intersection**:
- The intersection of the intervals \( \left( \frac{\pi}{4}, \frac{3\pi}{4} \right) \) and \( \left( -\frac{\pi}{4}, \frac{\pi}{4} \right) \) is empty, meaning this case does not provide a valid solution.

6. **Case 2: Both Points Below the Line**:
- For point \( P \):
\[ \sin \theta < \frac{1}{\sqrt{2}} \]
- For point \( Q \):
\[ \frac{1}{\sqrt{2}} > \cos \theta \]

7. **Analyzing the Inequalities**:
- The inequality \( \sin \theta < \frac{1}{\sqrt{2}} \) implies:
\[ \theta \in \left( -\frac{\pi}{4}, \frac{\pi}{4} \right) \]
- The inequality \( \cos \theta < \frac{1}{\sqrt{2}} \) implies:
\[ \theta \in \left( \frac{\pi}{4}, \frac{3\pi}{4} \right) \]

8. **Finding the Intersection**:
- The intersection of the intervals \( \left( -\frac{\pi}{4}, \frac{\pi}{4} \right) \) and \( \left( \frac{\pi}{4}, \frac{3\pi}{4} \right) \) is also empty.

9. **Final Interval**:
- The valid intervals for \( \theta \) that satisfy both conditions must be combined:
\[ \theta \in \left( -\frac{\pi}{4}, \frac{\pi}{4} \right) \cup \left( \frac{\pi}{4}, \frac{3\pi}{4} \right) \]
- Thus, the final answer is:
\[ \theta \in \left( -\frac{\pi}{4}, \frac{3\pi}{4} \right) \]