To solve the equation cos^2x - 5cos2x = 2sin^2x, we can use some trigonometric identities to simplify the equation.

First, we know that cos(2x) = cos^2x - sin^2x. Therefore, we can rewrite the equation as:

cos^2x - 5(cos^2x - sin^2x) = 2sin^2x
cos^2x - 5cos^2x + 5sin^2x = 2sin^2x
-4cos^2x + 5sin^2x = 2sin^2x
-4cos^2x + 5sin^2x - 2sin^2x = 0
-4cos^2x + 3sin^2x = 0

Now we can use the Pythagorean identity sin^2x + cos^2x = 1 to rewrite the equation in terms of sin^2x:

-4(1 - sin^2x) + 3sin^2x = 0
-4 + 4sin^2x + 3sin^2x = 0
7sin^2x - 4 = 0
7sin^2x = 4
sin^2x = 4/7
sinx = ±√(4/7)

Therefore, the solutions for the equation cos^2x - 5cos2x = 2sin^2x are x = arcsin(±√(4/7)) + 2πn, where n is an integer.