Решить уравнение (подробно)

2sin² x/2=cos (3π/2+x/2)

2sin² (x/2) = cos (3π/2 + x/2)

2sin ² (x/2) = cos (π + π/2 + x/2)

2sin² (x/2) = sin (x/2)

2sin² (x/2) - sin (x/2) = 0

sin (x/2) [2sin (x/2) - 1) = 0

1) sin (x/2) = 0

x/2 = πn, n ∈ Z

x = 2πn, n ∈ Z

2) 2sin (x/2) - 1 = 0

sin (x/2) = 1/2

x/2 = (-1) ⁿπ/6 + πk, k ∈ Z

x = (-1) ⁿπ/3 + 2πk, k ∈ Z

Ответ: x = 2πn, n ∈ Z; (-1) ⁿπ/3 + 2πk, k ∈ Z.