The function \( \tan(x) \) is discontinuous at specific points. These points occur when the cosine function \( \cos(x) \), which appears in the denominator of \( \tan(x) = \frac{\sin(x)}{\cos(x)} \), is equal to zero. 

Since \( \cos(x) = 0 \) at \( x = \frac{\pi}{2} + n\pi \) for any integer \( n \), the tangent function has vertical asymptotes at these points, causing discontinuities.

Therefore, \( \tan(x) \) is discontinuous at:

\[
x = \frac{\pi}{2} + n\pi \quad \text{for any integer } n.
\]

Would you like further details or any clarifications?

### Related Questions:
1. What is the behavior of \( \tan(x) \) near these discontinuities?
2. How do you find the points of discontinuity for any general trigonometric function?
3. Can you explain the difference between removable and non-removable discontinuities?
4. What are vertical asymptotes, and how are they related to discontinuities?
5. How does the periodicity of the tangent function affect its graph?

**Tip:** Always check the denominator when dealing with rational functions to find points of discontinuity.

Explore the points of discontinuity in the tangent function tan(x) by examining when cos(x) equals zero. This solution identifies specific points where tan(x) is undefined and highlights the relationship between tangent function properties, vertical asymptotes, and periodicity. Suitable for high school students.

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