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$1-co{s}^2(x)$ 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$x$ = $$-(-4)\pm \surd ((-4{)}^2-4(4)(1))$$/$2(4)$ cos$x$ = $$4\pm \surd (16-16)$$/$8$ cos$x$ = $$4\pm 0$$/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.