4cos^2x - 3sinx - 3 = 0  

4(1 - sin^2x) - 3sinx - 3 = 0

4 - 4sin^2x - 3sinx - 3 = 0  

- 4sin^2x - 3sinx + 1 = 0   // : (-1)

 4sin^2x + 3sinx - 1 = 0

Пусть  sinx = t , тогда:

4t^2 + 3t - 1 = 0  

Δ = 9 + 16 = 25

t1 = ( - 3 +5)/8 = 1/4  

t2 = ( - 3 - 5)/8 = - 1  

sinx =  1/4

x = (-1)^k arcsin(1/4) + pik, k ∈ Z

sinx = 1  

x = -π/2 + 2πn, n ∈ Z.