To simplify this trigonometric expression, we can use the half-angle and double-angle identities, along with the Pythagorean identity.

The given expression is:

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

Using the half-angle identity for cosine, we have:

cos^2(x/2) = 1/2 (1 + cos(x))

Now, let's simplify the expression with the half-angle identity:

1/2 (1 + cos(x)) - sin^2(2x) = 1

Now, we need to express sin^2(2x) in terms of cosines using the double-angle identity:

sin^2(2x) = 1 - cos^2(2x)

Substitute sin^2(2x) = 1 - cos^2(2x) into the expression:

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

Expand the expression:

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

Combine the constants:

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

Rearrange the terms:

1/2 + cos(x) + cos^2(2x) = 1

Now, we can simplify this expression further by using the Pythagorean identity:

cos^2(2x) = 1 - sin^2(2x)

Substitute cos^2(2x) = 1 - sin^2(2x) into the expression:

1/2 + cos(x) + 1 - sin^2(2x) = 1

Combine the constants:

3/2 + cos(x) - sin^2(2x) = 1

Now, rearrange the terms, and we have:

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

Therefore, the simplified expression is:

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