(Solution) - The Chebyshev polynomials a Prove that Tn t cos n arccos t n -(2025 Original AI-Free Solution)

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The Chebyshev polynomials:
(a) Prove that
Tn(t) = cos(n arccos t), n = 0, 1, 2,...
form a system of orthogonal polynomials under the weighted inner product

(b) What is || Tn ||?
(c) Write out the formulae for T0(t),.... T6(t) and plot their graphs.