Example 2 Find the values of the following expressions:
(1) $\sec 50^{\circ}+\operatorname{tg} 10$;
(2) $\cos \frac{2 \pi}{7}+\cos \frac{4 \pi}{7}+\cos \frac{6 \pi}{7}$;
(3) $\operatorname{tg} 6^{\circ} \operatorname{tg} 42^{\circ} \operatorname{tg} 66^{\circ} \operatorname{tg} 78^{\circ}$.
(1) Original expression $=\frac{1}{\cos 50^{\circ}}+\frac{\sin 10^{\circ}}{\cos 10^{\circ}}=\frac{2 \sin 50^{\circ}}{\sin 100^{\circ}}+\frac{\sin 10^{\circ}}{\cos 10^{\circ}}$
$$
=\frac{\sin 50^{\circ}+2 \sin 30^{\circ} \cos 20^{\circ}}{\cos 10^{\circ}}=\frac{2 \cos 30^{\circ} \cos 10^{\circ}}{\cos 10^{\circ}}=\sqrt{3} ;
$$
(2) Original expression $=\frac{2 \sin \frac{2 \pi}{7}}{2 \sin \frac{2 \pi}{7}} \cdot\left(\cos \frac{2 \pi}{7}+\cos \frac{4 \pi}{7}+\cos \frac{6 \pi}{7}\right)$
$$
\begin{array}{l}
=\frac{1}{2 \sin \frac{2 \pi}{7}} \cdot\left[\sin \frac{4 \pi}{7}+\sin \frac{6 \pi}{7}-\sin \frac{2 \pi}{7}+\sin \frac{8 \pi}{7}-\sin \frac{4 \pi}{7}\right] \\
=-\frac{1}{2} .
\end{array}
$$
(3) Since $\sin 6^{\circ} \sin 42^{\circ} \sin 66^{\circ} \sin 78^{\circ}$
$$
\begin{array}{l}
=\sin 6^{\circ} \cos 12^{\circ} \cos 24^{\circ} \cos 48^{\circ} \\
=\frac{2 \cos 6^{\circ} \sin 6^{\circ} \sin 12^{\circ} \sin 24^{\circ} \sin 48^{\circ}}{2 \cos 6^{\circ}}=\frac{\sin 96^{\circ}}{16 \cos 6^{\circ}}=\frac{1}{16} \\
\cos 6^{\circ} \cos 42^{\circ} \cos 66^{\circ} \cos 78 \\
=\frac{1}{4}\left(\cos 72^{\circ}+\cos 60^{\circ}\right)\left(\cos 12^{\circ}+\cos 36^{\circ}\right) \\
=\frac{1}{4}\left(2 \sin 18^{\circ} \cos 36^{\circ}-\frac{1}{4}\right) \\
=\frac{1}{4}\left[\frac{2 \sin 18^{\circ} \cos 18^{\circ} \cos 36^{\circ}}{\cos 18^{\circ}}-\frac{1}{4}\right]=\frac{1}{16} .
\end{array}
$$

Therefore, the original expression $=1$.