To show that the function \( y = \tan(-2x) \) is periodic, we need to demonstrate that there exists a positive constant \( T \) such that:

\[
\tan(-2(x + T)) = \tan(-2x)
\]

### Step-by-Step Solution:

1. **Start with the standard tangent function's periodicity:**
   The tangent function \( \tan(x) \) is periodic with a fundamental period of \( \pi \), meaning:
   \[
   \tan(x + \pi) = \tan(x)
   \]

2. **For \( y = \tan(-2x) \):**
   We know that \( y = \tan(-2x) \) is simply a transformation of the standard tangent function. Specifically, the negative sign reflects the function horizontally, and the factor of 2 compresses it horizontally by a factor of 2.

3. **Find the period for \( y = \tan(-2x) \):**
   To find the period of this transformed function, we apply the transformation to the period of the standard tangent function:
   \[
   T = \frac{\pi}{|2|}
   \]
   So, the period is:
   \[
   T = \frac{\pi}{2}
   \]

4. **Verify the periodicity:**
   To verify this, let's check if adding the period \( T = \frac{\pi}{2} \) satisfies the periodicity condition for \( y = \tan(-2x) \):
   \[
   \tan(-2(x + T)) = \tan(-2(x + \frac{\pi}{2})) = \tan(-2x - \pi)
   \]
   Using the periodicity of the standard tangent function \( \tan(x + \pi) = \tan(x) \), we have:
   \[
   \tan(-2x - \pi) = \tan(-2x)
   \]
   Therefore, \( y = \tan(-2x) \) is periodic with period \( T = \frac{\pi}{2} \).

### Conclusion:
The function \( y = \tan(-2x) \) has a period of \( \frac{\pi}{2} \), and thus it is periodic.

---

Would you like further details or have any questions about this explanation?  
Here are some related questions:

1. What is the period of the function \( y = \tan(kx) \) for any constant \( k \)?
2. How does a horizontal compression affect the period of trigonometric functions?
3. How does the reflection of the argument \( -2x \) affect the tangent function?
4. What is the difference between periodicity and symmetry in trigonometric functions?
5. Can the period of \( y = \tan(-2x) \) change if other transformations are applied?

**Tip:** When working with periodic functions, always look for transformations that affect the period, such as horizontal stretches/compressions or shifts.

Explore how to prove that the function y = tan(-2x) is periodic, with a period of π/2. This detailed step-by-step solution covers key trigonometric concepts like periodicity and function transformations, suitable for high school students in Grades 10-12.

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