Consider the following statements
I. `cosec^(-1)(-(2)/(sqrt3))=-(pi)/(3)`
II. `sec^(-1)((2)/(sqrt3))=(pi)/(6)`
Which of the above statement(s) is//are correct?

Options

A. Only I
B. Only II
C. Both I and II
D. Neither I nor II

Answer

To determine the correctness of the given statements, we will analyze each statement step by step.

### Statement I: \( \csc^{-1}\left(-\frac{2}{\sqrt{3}}\right) = -\frac{\pi}{3} \)

1. **Understanding the Inverse Cosecant Function**:
- The cosecant function is defined as \( \csc(\theta) = \frac{1}{\sin(\theta)} \).
- Therefore, \( \csc^{-1}(x) \) gives us an angle \( \theta \) such that \( \csc(\theta) = x \).

2. **Setting Up the Equation**:
- Let \( \theta = \csc^{-1}\left(-\frac{2}{\sqrt{3}}\right) \).
- This means \( \csc(\theta) = -\frac{2}{\sqrt{3}} \).

3. **Finding Sine**:
- Since \( \csc(\theta) = \frac{1}{\sin(\theta)} \), we have:
\[ \sin(\theta) = -\frac{\sqrt{3}}{2} \]

4. **Finding the Angle**:
- The angle \( \theta \) for which \( \sin(\theta) = -\frac{\sqrt{3}}{2} \) is \( \theta = -\frac{\pi}{3} \) (in the fourth quadrant).

5. **Conclusion for Statement I**:
- Thus, \( \csc^{-1}\left(-\frac{2}{\sqrt{3}}\right) = -\frac{\pi}{3} \) is **correct**.

### Statement II: \( \sec^{-1}\left(\frac{2}{\sqrt{3}}\right) = \frac{\pi}{6} \)

1. **Understanding the Inverse Secant Function**:
- The secant function is defined as \( \sec(\phi) = \frac{1}{\cos(\phi)} \).
- Therefore, \( \sec^{-1}(x) \) gives us an angle \( \phi \) such that \( \sec(\phi) = x \).

2. **Setting Up the Equation**:
- Let \( \phi = \sec^{-1}\left(\frac{2}{\sqrt{3}}\right) \).
- This means \( \sec(\phi) = \frac{2}{\sqrt{3}} \).

3. **Finding Cosine**:
- Since \( \sec(\phi) = \frac{1}{\cos(\phi)} \), we have:
\[ \cos(\phi) = \frac{\sqrt{3}}{2} \]

4. **Finding the Angle**:
- The angle \( \phi \) for which \( \cos(\phi) = \frac{\sqrt{3}}{2} \) is \( \phi = \frac{\pi}{6} \) (in the first quadrant).

5. **Conclusion for Statement II**:
- Thus, \( \sec^{-1}\left(\frac{2}{\sqrt{3}}\right) = \frac{\pi}{6} \) is **correct**.

### Final Conclusion:
Both statements are correct.

### Answer:
Both statements I and II are correct.

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