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)

Given the function \(\psi(x) = \tan x\), let's evaluate the expressions step-by-step.

1. **Evaluate \(\psi\left(\frac{\pi}{6}\right)\):**
   \[
   \psi\left(\frac{\pi}{6}\right) = \tan\left(\frac{\pi}{6}\right)
   \]
   Using the known value:
   \[
   \tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}}
   \]
   Thus,
   \[
   \psi\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}}
   \]

2. **Evaluate \(\psi\left(x - \frac{\pi}{2}\right)\):**
   \[
   \psi\left(x - \frac{\pi}{2}\right) = \tan\left(x - \frac{\pi}{2}\right)
   \]
   Using the tangent identity \(\tan\left(x - \frac{\pi}{2}\right) = -\cot x\), we get:
   \[
   \psi\left(x - \frac{\pi}{2}\right) = -\cot x
   \]

3. **Evaluate \(\psi(-x)\):**
   \[
   \psi(-x) = \tan(-x)
   \]
   Using the odd function property of tangent, \(\tan(-x) = -\tan x\), we get:
   \[
   \psi(-x) = -\tan x
   \]

4. **Express \(\psi(2x)\) as a function of \(\psi(x)\):**
   We want to express \(\tan(2x)\) in terms of \(\tan(x)\). Using the double-angle identity for tangent:
   \[
   \tan(2x) = \frac{2\tan x}{1 - \tan^2 x}
   \]
   Since \(\psi(x) = \tan x\), let \(\psi(x) = t\). Then:
   \[
   \psi(2x) = \frac{2t}{1 - t^2}
   \]

So, summarizing the results:

1. \(\psi\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}}\)
2. \(\psi\left(x - \frac{\pi}{2}\right) = -\cot x\)
3. \(\psi(-x) = -\tan x\)
4. \(\psi(2x) = \frac{2\psi(x)}{1 - (\psi(x))^2}\)

Would you like more details on any of these steps or have any questions?

Here are 5 relative questions for further exploration:

1. How would you evaluate \(\psi\left(\frac{\pi}{4}\right)\)?
2. What is the value of \(\psi\left(\frac{\pi}{3}\right)\)?
3. How does the function \(\psi(x)\) behave as \(x\) approaches \(\frac{\pi}{2}\)?
4. Can you derive the identity for \(\psi(3x)\) in terms of \(\psi(x)\)?
5. What are the critical points of \(\psi(x) = \tan x\) within the interval \(0 \leq x < \pi\)?

**Tip:** When working with trigonometric functions, knowing and using fundamental identities can significantly simplify the problem-solving process.

Explore solutions for evaluating tan x, its variants like tan(2x), tan(x - π/2), and tan(-x) using trigonometric identities and properties. Detailed step-by-step explanations provided. Suitable for advanced high school students studying trigonometry.

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