Example 4 Find the smallest positive period of the function $y=\frac{2 \tan x}{1-\tan ^{2} x}$.

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Analysis: The conventional solution is to use the inverse operation of the universal formula to convert the function into $y=\frac{2 \tan x}{1-\tan ^{2} x}=\tan 2 x$. Without considering the domain factors, its smallest positive period is $\frac{\pi}{2}$.

However, if we delve deeper into the analysis, we can see that the domain of $y=\frac{2 \tan x}{1-\tan ^{2} x}=\tan 2 x$ is not an equivalent transformation. Is the smallest positive period of the original function still $\frac{\pi}{2}$? This requires us to determine the "smallest positive period" of the domain before and after the transformation of the original function. We can use the unit circle to represent its domain (empty points indicate non-domain).
The domain of $\frac{2 \tan x}{1-\tan ^{2} x}$ is $x \neq k \pi+\frac{\pi}{2}$ and $x \neq \frac{1}{2} k \pi+\frac{\pi}{4}(k \in \mathbf{Z})$, which is represented on the unit circle as shown in Figure 2-2-1, and its domain's "smallest positive period" is $\pi$; the domain of $\tan 2 x$ is $x \neq \frac{1}{2} k \pi+\frac{\pi}{4}(k \in \mathbf{Z})$, which is represented on the unit circle as shown in Figure 2-2-2, and its domain's "smallest positive period" is $\frac{\pi}{2}$.
The least common multiple of $\frac{\pi}{2}$, $\pi$, and $\frac{\pi}{2}$ is $\pi$, so the smallest positive period of the original function is $\pi$.
Note: Example 4 reminds us: (1) When using trigonometric transformations to find the smallest positive period of a trigonometric function, we must pay attention to the changes in the domain before and after the transformation. If the domain remains unchanged, we can directly find its smallest positive period; if the domain changes, then we use the unit circle method to find it; (2) When applying definitions, formulas, and theorems to solve problems, we must pay attention to their prerequisite conditions and implicit conditions to prevent non-equivalent transformations that can lead to misunderstandings.