Sin^2 (x) - 2sin (x) * cos (x) - 3cos^2 (x) + 2=0

1. Заменим свободный член уравнения удвоенной суммой квадратов sinx и cosx:

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

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

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

3sin^2 (x) - 2sin (x) cos (x) - cos^2 (x) = 0.

2. Разделим обе части уравнения на cos^2 (x):

3tg^2 (x) - 2tg (x) - 1 = 0;

D/4 = 1^2 + 3 * 1 = 1 + 3 = 4;

√ (D/4) = 2;

tgx = (1 ± 2) / 3;

a) tgx = (1 - 2) / 3 = - 1/3;

x = - arctg (1/3) + πk, k ∈ Z.

b) tgx = (1 + 2) / 3 = 3 / 3 = 1;

x = π/4 + πk, k ∈ Z.

Ответ: - arctg (1/3) + πk; π/4 + πk, k ∈ Z.