2cos^2 x+3sin2x-8sin^2x=0

2 сos²x + 3sin2x - 8sin²x = 0

8sin²x - 3sin2x - 2cos²x = 0

8sin²x - 6sinxcosx - 2cos²x = 0 |:2cos²x, cosx ≠ 0

4tg²x - 3tgx - 1 = 0

Пусть t = tgx.

4t² - 3t - 1 = 0

D = 9 + 4·4 = 25 = 5²

t₁ = (3 + 5) / 8 = 8/8 = 1

t₂ = (3 - 5) / 8 = - 2/8 = - 1/4

Обратная замена:

tgx = 1

x = π/4 + πn, n ∈ Z

tgx = - 1/4

x = arctg (-1/4) + πk, k ∈ Z

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