Problem - 2073
Let sequences {$a_n$} and {$b_n$} satisfy: $a_n=a_{n-1}\cos{\theta} - b_{n-1}\sin{\theta}$ and $b_n=a_{n-1}\sin{\theta}+b_{n-1}\cos{\theta}$. If $a_1=1$ and $b_1=\tan{\theta}$, where $\theta$ is a known real number, find the general formula for {$a_n$} and {$b_n$}.
L et $z_n=a_n+i\cdot b_n$, then $$\frac{z_n}{z_{n-1}}=\frac{(a_{n-1}\cos{\theta} - b_{n-1}\sin{\theta})+i\cdot(a_{n-1}\sin{\theta}+b_{n-1}\cos{\theta})}{a_{n-1}+i\cdot b_{n-1}}=\cos{\theta}+i\cdot\sin{\theta}$$
Hence $z_n=(1+i\cdot\tan{\theta})(\cos{\theta}+i\cdot\sin{\theta})^{n-1}=\sec{\theta}(\cos{\theta}+i\cdot\sin{\theta})^n=\sec{\theta}\cos{n\theta} + i\cdot\sin{n\theta}$ which leads to
$$a_n=\sec{\theta}\cos{n\theta},\quad\text{and}\quad b_n=\sec{\theta}\sin{n\theta}$$