Example 1 Find the smallest positive period $T$ of the function $f(x)=\frac{2 \sin x+1}{3 \sin x-5}$.
Solving: Since $\frac{2 \sin x+1}{3 \sin x-5}=\frac{2}{3}+\frac{13}{3(3 \sin x-5)}$,
thus $f(x+T)=f(x) \Leftrightarrow \frac{2}{3}+\frac{13}{3[3 \sin (x+T)-5]}=\frac{2}{3}+\frac{13}{3(3 \sin x-5)}$
$$
\Leftrightarrow \sin (x+T)=\sin x \text {. }
$$

From this, we know that the smallest positive period of $f(x)$ and $y=\sin x$ is $T=2 \pi$.
This method, although seemingly simple, can conveniently determine the smallest positive period of a large number of more complex functions. For example:
(1) The function $f(x)=\sqrt[2 n+1]{1-\sqrt[2 k+1]{1-\cos a x}}\left(n, k \in \mathbf{N}^{*}, a\right.$ is a non-zero real number $)$ has the smallest positive period of $\frac{2 \pi}{|a|}$.

Proof: From $f(x+T)=f(x) \Leftrightarrow \cos a(x+T)=\cos a x$, we know that the smallest positive period of $f(x)$ and the function $y=\cos a x$ is the same, both being $\frac{2 \pi}{|a|}$.
(2) The function $y=\sin ^{n} x(x \in \mathbf{R})$. When $n$ is even, the period is $k \pi(k \in \mathbf{Z}, k \neq 0)$, and the smallest positive period is $\pi$; when $n$ is odd, the period is $2 k \pi(k \in \mathbf{Z}, k \neq 0)$, and the smallest positive period is $2 \pi$.
Proof: It is easy to prove that $y=\sin ^{n} x(x \in \mathbf{R})$ is a periodic function (obviously $2 \pi$ is one of its periods).
Let $k(k \neq 0)$ be a period of $y=\sin ^{n} x(x \in \mathbf{R})$.
By the definition of a period, we have $\sin ^{n} x=\sin ^{n}(x+k)(x \in \mathbf{R})$.
When $n$ is odd, (1) holds if and only if $\sin x=\sin (x+k)(x \in \mathbf{R})$,
i.e., $k=2 m \pi(m \in \mathbf{Z}, m \neq 0)$. The smallest positive period is $2 \pi$.
Therefore, when $n$ is odd, the period of $y=\sin ^{n} x(x \in \mathbf{R})$ is $2 m \pi(m \in \mathbf{Z}, m \neq 0)$, and the smallest positive period is $2 \pi$.
When $n$ is even, (1) holds if and only if $|\sin x|=|\sin (x+k)|(x \in \mathbf{R})$.
Therefore, when $n$ is even, the period of $y=\sin ^{n} x(x \in \mathbf{R})$ is $m \pi(m \in \mathbf{Z}, m \neq 0)$, and the smallest positive period is $\pi$.
(3) The function $y=\cos ^{n} x(x \in \mathbf{R})$. When $n$ is even, the period is $k \pi(k \in \mathbf{Z}, k \neq 0)$, and the smallest positive period is $\pi$; when $n$ is odd, the period is $2 k \pi(k \in \mathbf{Z}, k \neq 0)$, and the smallest positive period is $2 \pi$.
This can be proven similarly to (2).
Generalization (1) The function $y=A \sin ^{n}(\omega x+\varphi)(x \in \mathbf{R}, A \omega \neq 0, A, \omega, \varphi$ are constants $)$.
When $n$ is odd, the period is $\frac{2 k \pi}{\omega}(k \in \mathbf{Z}, k \neq 0)$, and the smallest positive period is $\frac{2 \pi}{|\omega|}$;
When $n$ is even, the period is $\frac{k \pi}{\omega}(k \in \mathbf{Z}, k \neq 0)$, and the smallest positive period is $\frac{\pi}{|\omega|}$.
(2) The function $y=A \cos ^{n}(\omega x+\varphi)(x \in \mathbf{R}, A \cdot \omega \neq 0, A, \omega, \varphi$ are constants $)$.
When $n$ is odd, the period is $\frac{2 k \pi}{\omega}(k \in \mathbf{Z}, k \neq 0)$, and the smallest positive period is $\frac{2 \pi}{|\omega|}$;
When $n$ is even, the period is $\frac{k \pi}{\omega}(k \in \mathbf{Z}, k \neq 0)$, and the smallest positive period is $\frac{\pi}{|\omega|}$.