3. Let $n(n \geqslant 2)$ be a given integer, and $x_{1}, x_{2}, \cdots$, $x_{n}$ be real numbers. Then the maximum value of $\sin x_{1} \cdot \cos x_{2}+\sin x_{2} \cdot \cos x_{3}+$ $\cdots+\sin x_{n} \cdot \cos x_{1}$ is $\qquad$.
3. $\frac{n}{2}$.
$$
\begin{array}{l}
\sum_{i=1}^{n} \sin x_{i} \cdot \cos x_{i+1} \leqslant \frac{1}{2} \sum_{i=1}^{n}\left(\sin ^{2} x_{i}+\cos ^{2} x_{i+1}\right) \\
=\frac{1}{2} \sum_{i=1}^{n}\left(\sin ^{2} x_{i}+\cos ^{2} x_{i}\right)=\frac{n}{2} .
\end{array}
$$

where $x_{n+1}=x_{1}$.
When $x_{1}=x_{2}=\cdots=x_{n}=\frac{\pi}{4}$, the equality holds.
Therefore, the maximum value sought is $\frac{n}{2}$.