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Apply the cosine double-angle identity twice:This finite trigonometric sum is already the Fourier series. In increasing harmonic order, including the constant term, the first two nonzero terms are

Use the Taylor series $\cos x=1-x^2/2+O(x^4)$. Squaring gives ${\cos }^{2}x=1-x^2+O(x^4)$, and henceThe first two nonzero terms are therefore $x^3$ and $-x^5$.