`cosec"^(-1)(-sqrt(2))`

Explanation

To solve the problem \( \csc^{-1}(-\sqrt{2}) \), we will follow these steps:

### Step 1: Understand the definition of \( \csc^{-1}(x) \)
The cosecant inverse function, \( \csc^{-1}(x) \), is defined as the angle \( \theta \) such that \( \csc(\theta) = x \) and \( \theta \) is in the range \( (-\frac{\pi}{2}, 0) \cup (0, \frac{\pi}{2}) \).

### Step 2: Set up the equation
We need to find \( \theta \) such that:
\[
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