To solve the equation, we first need to isolate the cosine term and then solve for x.

1 - 2cos^2(2x) = √2/2

Subtract 1 from both sides:

-2cos^2(2x) = √2/2 - 1
-2cos^2(2x) = -√2/2

Divide by -2:

cos^2(2x) = √2/4

Take the square root of both sides:

cos(2x) = ±√2/2

Now, to solve for x, we need to consider the possible values of 2x.

For cos(2x) = √2/2:
2x = π/4 + 2πn or 2x = 7π/4 + 2πn
x = π/8 + πn or x = 7π/8 + πn, where n is an integer.

For cos(2x) = -√2/2:
2x = 3π/4 + 2πn or 2x = 5π/4 + 2πn
x = 3π/8 + πn or x = 5π/8 + πn, where n is an integer.

So, the solutions for the equation 1 - 2cos^2(2x) = √2/2 are x = π/8 + πn, x = 7π/8 + πn, x = 3π/8 + πn, and x = 5π/8 + πn, where n is an integer.