10. Given the sequences $\left\{a_{n}\right\},\left\{b_{n}\right\}$ satisfy $\left\{\begin{array}{l}a_{n}=a_{n-1} \cos \theta-b_{n-1} \sin \theta \\ b_{n}=a_{n-1} \sin \theta+b_{n-1} \cos \theta\end{array}\right.$, and $a_{1}=1, b_{1}=\tan \theta, \theta$ is a known real number, find the general terms of the sequences $\left\{a_{n}\right\},\left\{b_{n}\right\}$.
\begin{array}{l}\text { 10. Solution: Let } z_{n}=a_{n}+b_{n} \mathrm{i}, a_{n}, b_{n} \in \mathbf{R}, n \in \mathbf{N} \text {. } \\ \text { Then } \frac{z_{n}}{z_{n-1}}=\frac{\left(a_{n-1} \cos \theta-b_{n-1} \sin \theta\right)+\left(a_{n-1} \sin \theta+b_{n-1} \cos \theta\right) \mathrm{i}}{a_{n-1}+b_{n-1} \mathrm{i}}=\cos \theta+\mathrm{i} \sin \theta \text {, } \\ \therefore z_{n}=(1+\mathrm{itan} \theta)(\cos \theta+\mathrm{i} \sin \theta)^{n-1}=\sec \theta(\cos \theta+\mathrm{isin} \theta)^{n} \\ =\sec \theta \cos n \theta+\mathrm{isec} \theta \sin n \theta, \therefore a_{n}=\sec \theta \cdot \cos n \theta, b_{n}=\sec \theta \cdot \sin n \theta \text {. }\end{array}