The value of tan 6° tan 42° tan 66° tan 78° is

$\begin{array}{l}\text{ We have }\tan 6^{\circ }\tan {42}^{\circ }\tan {66}^{\circ }\tan {78}^{\circ }\\ =\tan 6^{\circ }\tan \left({60}^{\circ }-{18}^{\circ }\right)\tan \left({60}^{\circ }+6^{\circ }\right)\tan \left({60}^{\circ }+{18}^{\circ }\right)\\ =\frac{\tan 6^{\circ }\tan \left({60}^{\circ }+6^{\circ }\right)\tan {18}^{\circ }\tan \left({60}^{\circ }-{18}^{\circ }\right)\tan \left({60}^{\circ }+{18}^{\circ }\right)}{\tan {18}^{\circ }}=\frac{\tan 6^{\circ }\tan \left({60}^{\circ }+6^{\circ }\right)\tan \left(3\times {18}^{\circ }\right)}{\tan {18}^{\circ }}\\ =\frac{\tan 6^{\circ }\tan \left({60}^{\circ }-6\right)\tan \left({60}^{\circ }+6\right)}{\tan {18}^{\circ }}=\frac{\tan {18}^{\circ }}{\tan {18}^{\circ }}=1\end{array}$