Решить уравнение:

-sin (x/2) + cos (x/4) = 0

-sin (x/2) + cos (x/4) = 0

-2sin (x/4) cos (x/4) + cos (x/4) = 0

-cos (x/4) * (2sin (x/4) - 1) = 0

cosx/4=0 ⇒x/4=π/2+2πk⇒x=2π+8πk, k∈z

sinx/4=1/2

x/4=π/6+2πk, k∈z U x=5π/6+2πk, k∈z

x=2π/3+8πk, k∈z U x=10π/3+8πk, k∈z

-sin (x/2) + cos (x/4) = 0

Разложим sin (x/2) по формуле удвоенного аргумента

-2cos (x/4) sin (x/4) + cos (x/4) = 0

cos (x/4) [-2sin (x/4) + 1] = 0

cos (x/4) = 0

x/4 = π/2 + πn, n ∈ Z

x = 2π + 4πn, n ∈ Z

-2sin (x/4) = - 1

sin (x/4) = 1/2

x/4 = (-1) ⁿπ/6 + πk, k ∈ Z

x = (-1) ⁿ2π/3 + 4πk, k ∈ Z

Ответ: x = 2π + 4πn, n ∈ Z; (-1) ⁿ2π/3 + 4πk, k ∈ Z.