2cos^2x/2-3 sin (п-x/2) cos (2 п-x/2) + 7sin^2x/2=3

1. Преобразуем уравнение по формулам приведения:

2cos^2 (x/2) - 3sin (π - x/2) cos (2π - x/2) + 7sin^2 (x/2) = 3; 2cos^2 (x/2) - 3sin (x/2) cos (x/2) + 7sin^2 (x/2) = 3sin^2 (x/2) + 3cos^2 (x/2); 4sin^2 (x/2) - 3sin (x/2) cos (x/2) - cos^2 (x/2) = 0.

2. Разделим обе части на cos^2 (x/2):

4tg^2 (x/2) - 3tg (x/2) - 1 = 0;

D = 3^2 + 4 * 4 = 25;

tg (x/2) = (3 ± √25) / 8 = (3 ± 5) / 8;

1) tg (x/2) = (3 - 5) / 8 = - 2/8 = - 1/4;

x/2 = - arctg (1/4) + πk, k ∈ Z; x = - 2arctg (1/4) + 2πk, k ∈ Z;

2) tg (x/2) = (3 + 5) / 8 = 8/8 = 1;

x/2 = π/4 + πk, k ∈ Z; x = π/2 + 2πk, k ∈ Z.

Ответ: - 2arctg (1/4) + 2πk; π/2 + 2πk, k ∈ Z.