Applying the reduction formula twice, cos^4 x equals which expression?

1. Write cos^4 x as (cos^2 x)^2 and substitute cos^2 x = (1 + cos(2x)) / 2. 2. That gives ((1 + cos(2x)) / 2)^2, which expands to (1/4)(1 + 2 cos(2x) + cos^2(2x)). 3. Distributing gives 1/4 + (1/2) cos(2x) + (1/4) cos^2(2x). 4. Now apply the reduction formula again with the angle 2x, so cos^2(2x) = (1 + cos(4x)) / 2. 5. The last term becomes (1/4) times (1 + cos(4x)) / 2 = 1/8 + (1/8) cos(4x). 6. Adding the constants, 1/4 + 1/8 = 3/8. 7. So cos^4 x = 3/8 + (1/2) cos(2x) + (1/8) cos(4x). 8. A quick check at x = 0 gives 3/8 + 1/2 + 1/8 = 1, matching cos^4 0 = 1, which the other options fail. _Source: OpenStax Algebra and Trigonometry (CC BY 4.0), Ch 9 "Trigonometric Identities and Equations", section 9.3 Double-Angle, Half-Angle, and Reduction Formulas_