Example 1, find the values of $\cos 18^{\circ}$ and $\operatorname{tg} 18^{\circ}$.
Solve $\theta=18^{\circ}, \operatorname{ctg} 5 \theta=\operatorname{ctg} 90^{\circ}=0$.

Given $5 \operatorname{tg}^{4} 18^{\circ}-10 \operatorname{tg}^{2} 18^{\circ}+1=0$.
Since $\operatorname{tg} 18^{\circ}<1$, hence
$\operatorname{tg} 18^{\circ}=\frac{1}{5} \sqrt{25-10 \sqrt{5}}$.
$$
\begin{aligned}
\cos 18^{\circ}= & \sqrt{\frac{1}{1+\operatorname{tg}^{2} 18^{\circ}}} \\
& =\frac{1}{4} \sqrt{10+2 \sqrt{5}} .
\end{aligned}
$$