Step-by-Step Solution

1. Evaluate \(\tan^{-1}(-\frac{1}{\sqrt{3}})\):

   Since \(\tan(-\frac{\pi}{6}) = -\frac{1}{\sqrt{3}}\), we have \(\tan^{-1}(-\frac{1}{\sqrt{3}}) = -\frac{\pi}{6}\).

2. Evaluate \(\cot^{-1}(\frac{1}{\sqrt{3}})\):

   Since \(\cot(\frac{\pi}{3}) = \frac{1}{\sqrt{3}}\), we have \(\cot^{-1}(\frac{1}{\sqrt{3}}) = \frac{\pi}{3}\).

3. Evaluate \(\tan^{-1}[\sin(-\frac{\pi}{2})]\):

   Since \(\sin(-\frac{\pi}{2}) = -1\), we have \(\tan^{-1}[\sin(-\frac{\pi}{2})] = \tan^{-1}(-1)\). Since \(\tan(-\frac{\pi}{4}) = -1\), we have \(\tan^{-1}(-1) = -\frac{\pi}{4}\).

4. Combine the results:

   \(\tan^{-1}(-\frac{1}{\sqrt{3}})+\cot^{-1}(\frac{1}{\sqrt{3}})+\tan^{-1}[\sin(-\frac{\pi}{2})] = -\frac{\pi}{6} + \frac{\pi}{3} - \frac{\pi}{4}\)

5. Simplify the expression:

   To simplify, find a common denominator, which is 12.

   \(-\frac{\pi}{6} + \frac{\pi}{3} - \frac{\pi}{4} = -\frac{2\pi}{12} + \frac{4\pi}{12} - \frac{3\pi}{12} = \frac{-2\pi + 4\pi - 3\pi}{12} = \frac{-\pi}{12}\)