 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/ii/paper-2/41b/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

With

x = cos θ

T_{k} (x) = cos k θ

, proving orthogonality. In the convention

f \sim c_{0} + \sum_{k \geq 1} c_{k} T_{k}

c_{0} = π^{- 1} \int_{0}^{π} f (cos θ) d θ

and

c_{k} = 2 π^{- 1} \int_{0}^{π} f (cos θ) cos k θ d θ

. For

f = 1 - x^{2}

c_{0} = 2/ π

, odd coefficients vanish, and

c_{2 m} = - 4/ [π (4 m^{2} - 1)]

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