The value of `"cosec"^(-1)("cosec"(4pi)/3)` is

Options

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

Explanation

To solve the problem, we need to find the value of \( \csc^{-1}(\csc(4\pi/3)) \).

### Step-by-Step Solution:

1. **Identify the angle**:
We start with the angle \( \frac{4\pi}{3} \). This angle can be rewritten in terms of \( \pi \):
\[ \frac{4\pi}{3} = \pi + \frac{\pi}{3} \]

2. **Use the property of cosecant**:
The cosecant function has the property that:
\[ \csc(\pi + \theta) = -\csc(\theta) \]
Therefore, we can apply this property:
\[ \csc\left(\frac{4\pi}{3}\right) = \csc\left(\pi + \frac{\pi}{3}\right) = -\csc\left(\frac{\pi}{3}\right) \]

3. **Calculate \( \csc\left(\frac{\pi}{3}\right) \)**:
We know that:
\[ \csc\left(\frac{\pi}{3}\right) = \frac{1}{\sin\left(\frac{\pi}{3}\right)} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} \]
Thus, we have:
\[ \csc\left(\frac{4\pi}{3}\right) = -\frac{2}{\sqrt{3}} \]

4. **Substitute into the inverse cosecant**:
Now we substitute this value into the inverse cosecant function:
\[ \csc^{-1}\left(\csc\left(\frac{4\pi}{3}\right)\right) = \csc^{-1}\left(-\frac{2}{\sqrt{3}}\right) \]

5. **Determine the angle for the inverse cosecant**:
The value \( -\frac{2}{\sqrt{3}} \) corresponds to an angle in the fourth quadrant. The reference angle for \( \frac{2}{\sqrt{3}} \) is \( \frac{\pi}{3} \), thus:
\[ \csc^{-1}\left(-\frac{2}{\sqrt{3}}\right) = -\frac{\pi}{3} \]

### Final Answer:
Thus, the value of \( \csc^{-1}\left(\csc\left(\frac{4\pi}{3}\right)\right) \) is:
\[ \boxed{-\frac{\pi}{3}} \]