Решение
1+cos x+2 sin²x=0
1+cos x+2*(1 - cos²x) = 0
1 + cos x + 2 - 2cos²x = 0
2cos²x - cosx - 3 = 0
cosx = t, ItI ≤ 1
2t² - t - 3 = 0
D = 1 + 4*2*3 = 25
t₁ = (1 - 5)/4
t₁ = - 1
t₂ = (1 + 5)/4
t₂ = 6/4 > 1 не удовлетворяет условию ItI ≤ 1
cosx = - 1
x = π + 2πk, k ∈ Z