Дано: tg(a) + ctg(a) = 3

Используем тождество tg(a) * ctg(a) = 1:

(tg(a) + ctg(a))^2 = 3^2
tg^2(a) + 2 tg(a) ctg(a) + ctg^2(a) = 9
tg^2(a) + 2 + ctg^2(a) = 9
tg^2(a) + ctg^2(a) = 7

Далее найдем tg^3(a) + ctg^3(a):

tg^3(a) + ctg^3(a) = (tg(a) + ctg(a))(tg^2(a) - tg(a)ctg(a) + ctg^2(a))
tg^3(a) + ctg^3(a) = 3 (tg^2(a) - 1 + ctg^2(a))
tg^3(a) + ctg^3(a) = 3 (7 - 1)
tg^3(a) + ctg^3(a) = 18

Ответ: tg^3(a) + ctg^3(a) = 18.