2cos^2 * (π/2+x) + 3cosx=0

2 * cos^2 * (pi/2 + x) + 3 * cos x = 0;

2 * cos (pi/2 + x) * cos (pi/2 + x) + 3 * cos x = 0;

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

2 * sin x * sin x + 3 * cos x = 0;

2 * sin^2 x + 3 * cos x = 0;

2 * (1 - cos^2 x) + 3 * cos x = 0;

2 - 2 * cos^2 x + 3 * cos x = 0;

2 * cos^2 x - 3 * cos x - 2 = 0;

Найдем дискриминант уравнения.

D = 9 - 4 * 2 * (-2) = 25;

cos x = (3 + 5) / 4 = 8/4 = 2;

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

1) cos x = 2;

Уравнение не имеет корней.

2) cos x = - 1/2;

x = + -arccos (-1/2) + 2 * pi * n, n ∈ Z;

x = + -2 * pi/3 + 2 * pi * n, n ∈ Z.