2 cos² x - 1/2 sin 4x = 1;

Use the identity cos²x = (1 + cos 2x)/2 and sin 4x = 2 sin 2x cos 2x. The equation becomes 2 * (1 + cos 2x)/2 - 1/2 * (2 sin 2x cos 2x) = 1, which simplifies to 1 + cos 2x - sin 2x cos 2x = 1. This further simplifies to cos 2x - sin 2x cos 2x = 0. Factor out cos 2x: cos 2x (1 - sin 2x) = 0. This gives two possibilities: cos 2x = 0 or 1 - sin 2x = 0. If cos 2x = 0, then 2x = π/2 + nπ, so x = π/4 + nπ/2. If 1 - sin 2x = 0, then sin 2x = 1, so 2x = π/2 + 2nπ, which means x = π/4 + nπ. The solutions are x = π/4 + nπ/2.