 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2017/paper-2/5/solution

Apply the cosine double-angle identity twice:

cos^{4} x = (\frac{1 + c o s 2 x}{2})^{2} = \frac{3}{8} + \frac{1}{2} cos 2 x + \frac{1}{8} cos 4 x .

This finite trigonometric sum is already the Fourier series. In increasing harmonic order, including the constant term, the first two nonzero terms are

\frac{3}{8} + \frac{1}{2} cos 2 x .

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