Для начала найдем значение arctg(1/3):

arctg(1/3) = α

тогда tg(α) = 1/3

Далее известно, что tg^2(α) + 1 = 1/cos^2(α)

cos^2(α) = 1 / (tg^2(α) + 1)

cos(α) = ± sqrt(1 / (tg^2(α) + 1))

При tg(α) = 1/3, cos(α) = sqrt(1 / (1/9 + 1)) = sqrt(1 / (12/9)) = sqrt(9/12) = 3/2

Так как cos(α) > 0, то cos(α) = 3/2

tg(α) = 1/3 = sin(α)/cos(α) = sin(α) / (3/2) => sin(α) = 2/3

Используем формулу sin^2(α) + cos^2(α) = 1:

(2/3)^2 + (3/2)^2 = 1
4/9 + 9/4 = 1
(16/36) + (81/36) = 1
97/36 = 1
97 = 36

Теперь найдем tg^2(α):

tg^2(α) = sin^2(α) / cos^2(α) = (2/3)^2 / (3/2)^2 = 4/9 / 9/4 = 4/9 * 4/9 = 16/81

Итак, tg^2(arctg(1/3)) = 16/81.