 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2025/paper-2/4/solution

f (x) = \frac{1}{2} cos (2 x) + \frac{1}{2} cos (8 x) .

In the convention

f (x) = a_{0} /2 + \sum_{n \geq 1} (a_{n} cos n x + b_{n} sin n x)

, the only nonzero Fourier coefficients are

a_{2} = a_{8} = 1/2

. This is a finite trigonometric polynomial, and the identity holds at both endpoints as well.

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