If `[-sintheta]y=0` and `[cottheta]x+y=0` where `[]` denotes greatest integar function. Then which of the following is correct

Options

A. Infinite solution is `((pi)/(2),(2pi)/(3))` and a unique solution in `(pi,(7pi)/(6))`
B. Unique solution in `((pi)/(2),(2pi)/(3))` and infinite solutions in `(pi,(7pi)/(6))`
C. Unique solution is `((pi)/(2),(2pi)/(3))` and unique solution in `(pi,(7pi)/(6))`
D. Infinite solution in `((pi)/(2),(2pi)/(3))` and infinite solutions in `(pi,(7pi)/(6))`

Answer

To solve the problem, we need to analyze the given equations involving the greatest integer function (also known as the floor function) for the two intervals provided.

### Step 1: Analyze the first interval \( \theta \in \left(\frac{\pi}{2}, \frac{2\pi}{3}\right) \)

1. **Determine the range of \(-\sin \theta\)**:
- In the interval \( \left(\frac{\pi}{2}, \frac{2\pi}{3}\right) \), \(\sin \theta\) decreases from \(1\) to \(\frac{\sqrt{3}}{2}\).
- Thus, \(-\sin \theta\) will range from \(-1\) to \(-\frac{\sqrt{3}}{2}\).

2. **Find the greatest integer of \(-\sin \theta\)**:
- The greatest integer less than or equal to \(-\sin \theta\) in this range is \(-1\).
- Therefore, we have:
\[ \lfloor -\sin \theta \rfloor = -1 \]

3. **Analyze the cotangent function**:
- The cotangent function \(\cot \theta\) will range from \(0\) to \(-\frac{1}{\sqrt{3}}\).
- Thus, the greatest integer of \(\cot \theta\) in this range is also \(-1\).
- Therefore, we have:
\[ \lfloor \cot \theta \rfloor = -1 \]

4. **Set up the equations**:
- From the greatest integer function, we have:
\[ -(-1)y = 0 \implies y = 0 \]
- And for the cotangent equation:
\[ -1 \cdot x + y = 0 \implies -x + 0 = 0 \implies x = 0 \]

5. **Conclusion for the first interval**:
- The unique solution is \(x = 0\) and \(y = 0\).

### Step 2: Analyze the second interval \( \theta \in \left(\pi, \frac{7\pi}{6}\right) \)

1. **Determine the range of \(-\sin \theta\)**:
- In the interval \( \left(\pi, \frac{7\pi}{6}\right) \), \(\sin \theta\) ranges from \(0\) to \(-\frac{1}{2}\).
- Thus, \(-\sin \theta\) will range from \(0\) to \(\frac{1}{2}\).

2. **Find the greatest integer of \(-\sin \theta\)**:
- The greatest integer less than or equal to \(-\sin \theta\) in this range is \(0\).
- Therefore, we have:
\[ \lfloor -\sin \theta \rfloor = 0 \]

3. **Analyze the cotangent function**:
- The cotangent function \(\cot \theta\) will range from \(-\infty\) to \(-\sqrt{3}\).
- Thus, the greatest integer of \(\cot \theta\) can take any integer value starting from \(-\infty\).

4. **Set up the equations**:
- From the greatest integer function, we have:
\[ 0 \cdot y = 0 \implies y \text{ can be any value} \]
- And for the cotangent equation:
\[ \lfloor \cot \theta \rfloor \cdot x + y = 0 \]
- This means \(y = -\lfloor \cot \theta \rfloor \cdot x\), which implies \(y\) can take infinitely many values depending on \(x\).

5. **Conclusion for the second interval**:
- There are infinitely many solutions since \(y\) can be any value.

### Final Conclusion:
- For the interval \( \left(\frac{\pi}{2}, \frac{2\pi}{3}\right) \), there is a unique solution: \( (x, y) = (0, 0) \).
- For the interval \( \left(\pi, \frac{7\pi}{6}\right) \), there are infinitely many solutions.

### Answer:
- The correct option is **B**: Unique solution in \( \theta \in \left(\frac{\pi}{2}, \frac{2\pi}{3}\right) \) and infinite solutions in \( \theta \in \left(\pi, \frac{7\pi}{6}\right) \).