x+y=пи/4 tgx+tg (-y)=1/6решите систему уравнений

   1. Преобразуем:

• x + y = /4;
  tgx + tg(-y) = 1/6;
• y = /4 - x;
  tgx + tg(x - /4) = 1/6;
• y = /4 - x;
  tgx + (tgx - tg(/4))/(1 + tgx * tg(/4)) = 1/6;
• y = /4 - x;
  tgx + (tgx - 1)/(1 + tgx) = 1/6.

   2. Обозначим:

• tgx = t;

• t + (t - 1)/(1 + t) = 1/6;
• 6t(t + 1) + 6(t - 1) - (t + 1) = 0;
• 6t^2 + 6t + 6t - 6 - t - 1 = 0;
• 6t^2 + 11t - 7 = 0;

• D = 11^2 + 4 * 6 * 7 = 121 + 168 = 289;
• t = (-11 289)/12 = (-11 17)/12;

   1) t = (-11 - 17)/12 = -28/12 = -7/3;

• tgx = -7/3;
• x = -arctg(7/3) + k, k Z;

   2) t = (-11 + 17)/12 = 6/12 = 1/2;

• tgx = 1/2;
• x = arctg(1/2) + k, k Z.

   Ответ: -arctg(7/3) + k; arctg(1/2) + k, k Z.