Final answer:
The 28th derivative of f(x) = cos²(x) can be determined by expressing cos²(x) as (1 + cos(2x))/2 and using the periodic properties of trigonometric functions, leading to a conclusion that the 28th derivative is the same as the original function.
Step-by-step explanation:
The 28ᵗʰ derivative of the function f(x) = cos2(x) can be found using a known trigonometric identity and the periodic properties of trigonometric functions. The identity cos2(x) = (1 + cos(2x))/2 can be applied to express the original function in a form that makes differentiation more straightforward. Taking derivatives of trigonometric functions like cos(2x) repeatedly will yield a pattern that repeats every four derivatives since the derivatives of sine and cosine functions are cyclic with a period of four. For a high order like the 28th derivative, we can use this cyclic property to deduce that the 28ᵗʰ derivative is the same as the 4th derivative due to the pattern repeating every four derivatives. Therefore, applying these concepts to f(x) and taking into account the derivative pattern cos(x) → -sin(x) → -cos(x) → sin(x) → cos(x), the 28ᵗʰ derivative of f(x) is expected to be the same as the original function f(x).