Cos²х-sin²x-2cos²2x=0; Cos²x-½ sin2x=0

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

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

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

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

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

For cosx = 0, x = π/2

For cosx = sinx, we square both sides
cos²x = sin²
1 - sin²x = sin²
1 = 2sin²
sin²x = 1/
sinx = ±sqrt(1/2
x = π/4, 3π/4