sin^2x - 9sinx cosx + 3 cos^2x = - 1

sin^2 (x) - 9sin (x) cos (x) + 3 cos^2 (x) = - 1;

sin^2 (x) - 9sin (x) cos (x) + 3 cos^2 (x) + sin^2 (x) + cos^2 (x) = 0;

2sin^2 (x) - 9sin (x) cos (x) + 4cos^2 (x) = 0;

Разделим на cos^2 (x) ≠ 0;

2tg^2 (x) - 9 tg (x) + 4 = 0;

Сделаем замену переменной:

tg (x) = y;

2y^2 - 9y + 4 = 0;

D = 49;

y1 = 4; y2 = 1/2.

Сделаем обратную замену:

1) tg (x) = 4 = => x = arctg (4) + пk, k∈Z;

2) tg (x) = 1/2 = => x = arctg (1/2) + пn, n∈Z.

Ответ: x = arctg (4) + пk, k∈Z; x = arctg (1/2) + пn, n∈Z.