Evaluate : `"cosec"^(-1){"cosec" ((5pi)/(4))}`.

Answer

To evaluate the expression \( \csc^{-1}(\csc(5\pi/4)) \), we can follow these steps:

### Step 1: Understand the function
The cosecant function, \( \csc(x) \), is defined as \( \csc(x) = \frac{1}{\sin(x)} \). The inverse cosecant function, \( \csc^{-1}(x) \), gives us an angle whose cosecant is \( x \).

### Step 2: Calculate \( \csc(5\pi/4) \)
First, we need to find the value of \( \csc(5\pi/4) \):
\[ \csc(5\pi/4) = \frac{1}{\sin(5\pi/4)} \]
The angle \( 5\pi/4 \) is located in the third quadrant, where the sine function is negative. The reference angle for \( 5\pi/4 \) is \( \pi/4 \), and since \( \sin(\pi/4) = \frac{\sqrt{2}}{2} \), we have:
\[ \sin(5\pi/4) = -\frac{\sqrt{2}}{2} \]
Thus,
\[ \csc(5\pi/4) = \frac{1}{-\frac{\sqrt{2}}{2}} = -\frac{2}{\sqrt{2}} = -\sqrt{2} \]

### Step 3: Evaluate \( \csc^{-1}(-\sqrt{2}) \)
Now we need to find \( \csc^{-1}(-\sqrt{2}) \). The inverse cosecant function gives us an angle whose cosecant is \( -\sqrt{2} \). The angle corresponding to \( \csc(x) = -\sqrt{2} \) is:
\[ x = -\frac{\pi}{4} \quad \text{(in the range of } [-\frac{\pi}{2}, \frac{\pi}{2}] \text{)} \]
However, since \( 5\pi/4 \) is in the third quadrant, we should consider the principal value:
\[ \csc^{-1}(-\sqrt{2}) = -\frac{\pi}{4} \]

### Step 4: Final Result
Thus, we conclude that:
\[ \csc^{-1}(\csc(5\pi/4)) = 5\pi/4 \]
However, since \( \csc^{-1} \) returns values in the range \( [-\frac{\pi}{2}, \frac{\pi}{2}] \), we should express the final answer in terms of the principal value:
\[ \text{Final Answer: } -\frac{3\pi}{4} \]