Find the principal value of the following:
(i) `cosec^(-1)(2)` (ii) `tan^(-1) (-sqrt3)` (iii) `cos^(-1) (-(1)/(sqrt2))`

Explanation

To find the principal values of the given inverse trigonometric functions, we will solve each part step by step.

### (i) Find `cosec^(-1)(2)`

1. **Set the equation**: Let \( y = \csc^{-1}(2) \). This implies that \( \csc(y) = 2 \).
2. **Convert to sine**: Since \( \csc(y) = \frac{1}{\sin(y)} \), we have \( \sin(y) = \frac{1}{2} \).
3. **Find the angle**: The angle \( y \) for which \( \sin(y) = \frac{1}{2} \) is \( y = \frac{\pi}{6} \).
4. **Check the range**: The range of \( \csc^{-1}(x) \) is \( (-\frac{\pi}{2}, 0) \cup (0, \frac{\pi}{2}) \). Since \( \frac{\pi}{6} \) is in the range, we conclude that:
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