To find the solutions to these equations, we can use the double angle identities for cosine and sine.

For the equation cos²x - sin²x - 2cos²2x = 0:

Using the double angle identity for cosine, cos(2x) = 2cos²x - 1, we can rewrite the equation as:

cos²x - sin²x - 4cos²x + 2 = 0
-3cos²x - sin²x + 2 = 0
-3(1 - sin²x) - sin²x + 2 = 0
-3 + 3sin²x - sin²x + 2 = 0
2sin²x - 1 = 0
sin²x = 1/2
sinx = ±sqrt(1/2)
x = π/4, 3π/4, 5π/4, 7π/4

For the equation cos²x - ½sin2x = 0:

Using the double angle identity for sine, sin(2x) = 2sinxcosx, we can rewrite the equation as:

cos²x - sinxcosx = 0
cosx(cosx - sinx) = 0
cosx = 0 or cosx = sinx

For cosx = 0, x = π/2

For cosx = sinx, we square both sides:
cos²x = sin²x
1 - sin²x = sin²x
1 = 2sin²x
sin²x = 1/2
sinx = ±sqrt(1/2)
x = π/4, 3π/4