To solve the equation 5 - 4sin^2(x) = 4cos(x), we can use the Pythagorean identity sin^2(x) + cos^2(x) = 1.

Rearrange the given equation to have everything in terms of sin(x) or cos(x):
5 - 4sin^2(x) = 4cos(x)
4cos(x) = 5 - 4sin^2(x)

Replace sin^2(x) with 1 - cos^2(x) using the Pythagorean identity:
4cos(x) = 5 - 4(1 - cos^2(x))
4cos(x) = 5 - 4 + 4cos^2(x)
4cos(x) = 1 + 4cos^2(x)

Rearrange the equation in standard quadratic form:
4cos^2(x) - 4cos(x) + 1 = 0

This is a quadratic equation in terms of cos(x). We can solve it by using the quadratic formula:
cos(x) = [-(-4) ± √((-4)^2 - 4(4)(1))]/(2(4))
cos(x) = [4 ± √(16 - 16)]/(8)
cos(x) = [4 ± 0]/8
cos(x) = 1/2 or 1/2

Since cos(x) = 1/2, the possible values for x are x = π/3 or x = 5π/3.

Therefore, the solutions to the equation 5 - 4sin^2(x) = 4cos(x) are x = π/3 and x = 5π/3.