What is the value of the following functions (using principal value)
`"cosec"^(-1)("cosec"(3pi)/(4))`

Answer

To find the principal value of the expression \( \csc^{-1}(\csc(3\pi/4)) \), we can follow these steps:

### Step 1: Understand the expression
We need to evaluate \( \csc^{-1}(\csc(3\pi/4)) \). The cosecant function is the reciprocal of the sine function, so we can rewrite the expression as:
\[ \csc^{-1}(\csc(3\pi/4)) = \csc^{-1}\left(\frac{1}{\sin(3\pi/4)}\right) \]

### Step 2: Calculate \( \sin(3\pi/4) \)
The angle \( 3\pi/4 \) is in the second quadrant. In the second quadrant, the sine function is positive. The sine of \( 3\pi/4 \) can be calculated as:
\[ \sin(3\pi/4) = \sin\left(\pi - \frac{\pi}{4}\right) = \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \]

### Step 3: Calculate \( \csc(3\pi/4) \)
Now, we can find \( \csc(3\pi/4) \):
\[ \csc(3\pi/4) = \frac{1}{\sin(3\pi/4)} = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2} \]

### Step 4: Substitute back into the inverse cosecant function
Now we substitute back into the inverse cosecant function:
\[ \csc^{-1}(\csc(3\pi/4)) = \csc^{-1}(\sqrt{2}) \]

### Step 5: Find the principal value of \( \csc^{-1}(\sqrt{2}) \)
The principal value of \( \csc^{-1}(x) \) is defined for \( x \geq 1 \) and is in the range \( [-\frac{\pi}{2}, \frac{\pi}{2}] \) excluding zero. Since \( \sqrt{2} \) is positive and greater than 1, we can find:
\[ \csc^{-1}(\sqrt{2}) = \frac{\pi}{4} \]

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

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