Example 8. Calculate: $\operatorname{tg} 5^{\circ}+\operatorname{ctg} 5^{\circ}-2 \sec 80^{\circ}$. (79 National College Entrance Examination Supplementary Question)
$$
\begin{aligned}
\text { Original expression } & =\frac{2}{\sin \left(2 \times 5^{\circ}\right)}-\frac{2}{\cos 80^{\circ}} \\
& =\frac{2}{\sin 10^{\circ}}-\frac{2}{\sin 10^{\circ}}=0 .
\end{aligned}
$$