Given that $\psi(x) = \tan(x)$, let's solve each part of the problem.

1. $\psi\left(\frac{\pi}{6}\right)$

$\psi\left(\frac{\pi}{6}\right) = \tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}}$

2. $\psi\left(x - \frac{\pi}{2}\right)$

Using the trigonometric identity $\tan(x - \frac{\pi}{2}) = -\cot(x)$: $\psi\left(x - \frac{\pi}{2}\right) = \tan\left(x - \frac{\pi}{2}\right) = -\cot(x)$

3. $\psi(-x)$

Using the odd property of the tangent function $\tan(-x) = -\tan(x)$: [ \psi(-x) = \tan(-