Try this without calculator, it is really easy.

$\tan { 20 } \tan { 40 } \tan { 60 } \tan { 80 } can\quad be\quad written\quad as\\ \tan { 20 } \tan { (60-20) } \tan { (60+20) } \tan { 60 } \\ Now\\ \tan { 3\phi } =\tan { \phi } \tan { (60-\phi ) } \tan { (60+\phi ) } \\ \therefore \quad \tan { 20 } \tan { (60-20) } \tan { (60+20) } =\quad \tan { 60 } \\ \therefore \tan { 20 } \tan { 40 } \tan { 60 } \tan { 80 } \quad =\quad \tan { 60 } \quad \tan { 60 } =\sqrt { 3 } \sqrt { 3 } =3.$