To solve this equation, first we can use the identity 4cos^2x = 4(1 - sin^2x) to rewrite the equation:

4(1 - sin^2x) + 4sinx = 5
4 - 4sin^2x + 4sinx = 5

Rearranging terms, we get a quadratic equation:

-4sin^2x + 4sinx - 1 = 0

Now we can use the quadratic formula to solve for sinx:

sinx = [-4 ± sqrt((4)^2 - 4(-4)(-1))] / (2(-4))

sinx = [-4 ± sqrt(16 - 16)] / -8
sinx = [-4 ± 0] / -8
sinx = -4 / (-8)
sinx = 0.5

Therefore, sinx = 0.5

Now, to solve for cosx, we can substitute sinx = 0.5 back into the original equation:

4cos^2x + 2 = 5
4cos^2x = 3
cos^2x = 3/4
cosx = ±√(3/4) = ±√3 / 2

Therefore, the solutions to the equation are:
x = arcsin(0.5) or x = π - arcsin(0.5)
cosx = ±√3 / 2