tg (x+pi/4) + tg (x - pi/4) = tg x

1. Преобразуем уравнение:

tg (a + b) = (tga + tgb) / (1 - tga * tgb); tg (a - b) = (tga - tgb) / (1 + tga * tgb); tg (x + π/4) + tg (x - π/4) = tgx; (tgx + tg (π/4)) / (1 - tgx * tg (π/4)) + (tgx - tg (π/4)) / (1 + tgx * tg (π/4)) = tgx; (tgx + 1) / (1 - tgx) + (tgx - 1) / (1 + tgx) = tgx; ((tgx + 1) ^2 - (tgx - 1) ^2) / ((1 + tgx) (1 - tgx)) = tgx; (tg^2x + 2tgx + 1 - tg^2x + 2tgx - 1) / (1 - tg^2x) = tgx; 4tgx / (1 - tg^2x) - tgx = 0; tgx / (4 / (1 - tg^2x) - 1) = 0.

2. Приравняем множители к нулю:

[tgx = 0;

[4 / (1 - tg^2x) - 1 = 0; [tgx = 0;

[1 - tg^2x = 4; [tgx = 0;

[tg^2x = - 3 - нет решений; x = πk, k ∈ Z.

Ответ: πk, k ∈ Z.