Question: The Chebyshev polynomials \(T_{0}(x), T_{1}(x), T_{2}(x), \ldots\) are defined by the relation

At first, it's not even clear these are polynomials.

1. Use trigonometric identities to show that \(T_{0}(x)=1, T_{1}(x)=x\), and for all \(n \geq 1\),
   \[T_{n+1}(x)=2 x T_{n}(x)-T_{n-1}(x) .\]
   Use this to explain why all \(T_{n}\) are polynomials.
2. Show that the Chebyshev polynomials are orthogonal in the following sense:
   \[\int_{-1}^{1} T_{n}(x) T_{m}(x) \frac{d x}{\sqrt{1-x^{2}}}=0 \quad(n \neq m)\]
3. Assuming that a "Chebyshev expansion" exists:
   \[f(x)=\sum_{n=0}^{\infty} a_{n} T_{n}(x)\]
   find a formula for \(a_{n}\) in terms of \(f\) and \(n\).
4. Compute the Chebyshev series for \(f(x)=\arcsin x\).

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