If psi(x) = tan x find psi(pi / 6) psi(x - 1/2 * pi) psi(- x) and express psi(2x) as a function of psi(x) pm

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(-

Discover how to compute values of \( \tan(x) \) for specific angles like \( \frac{\pi}{6} \), evaluate expressions involving \( \tan(x) \), and apply trigonometric identities. Suitable for advanced high school students.

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