Let $[x]$ denote the greatest integer not exceeding $x$, and $\{x\}$ denote the fractional part of $x$ (i.e., $\{x\} = x - [x]$). Find the solution set of the equation $\cot [x] \cdot \cot \{x\} = 1$.

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Since $0 \leqslant\{x\}\cot 1>0$, that is $\cot \{x\} \neq 0$. Therefore, $\cot [x]=$ $\frac{1}{\cot \{x\}}=\tan \{x\}=\cot \left(\frac{\pi}{2}-\{x\}\right)$, hence $[x]=k \pi+\frac{\pi}{2}-\{x\}$, which means $[x]+\{x\}=k \pi+$ $\frac{\pi}{2}(k \in \mathbf{Z})$, that is $x=k \pi+\frac{\pi}{2}(k \in \mathbf{Z})$.