Используем формулу:
tg$2\alpha$ = $2tg\alpha$ / $1-t{g}^2(\alpha )$

Подставляем значения:
tg$2<em>20$ = $2tg20$ / $1-t{g}^2(20)$ = $2</em>tg20$ / $1-t{g}^2(20)$ tg$2<em>40$ = $2tg40$ / $1-t{g}^2(40)$ = $2</em>tg40$ / $1-t{g}^2(40)$ tg$2<em>60$ = $2tg60$ / $1-t{g}^2(60)$ = $2</em>tg60$ / $1-t{g}^2(60)$ tg$2<em>80$ = $2tg80$ / $1-t{g}^2(80)$ = $2</em>tg80$ / $1-t{g}^2(80)$

Получаем:
tg$40$ = tg$2<em>20$ tg$80$ = tg$2</em>40$ tg$120$ = tg$2<em>60$ tg$160$ = tg$2</em>80$

Итак, tg20tg40tg60tg80 = tg$40$ tg$80$ tg$120$ tg$160$ = tg$2<em>20$ tg$2<em>40$ tg$2<em>60$ tg$2<em>80$ = $2</em>tg20/(1-t{g}^2(20))$ $2</em>tg40/(1-t{g}^2(40))$ $2</em>tg60/(1-t{g}^2(60))$ $2</em>tg80/(1-t{g}^2(80))$.

Подставляем полученные значения и упрощаем выражение.