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The following exercise appears in Ridgway Scott's Numerical Analysis:

ex

Where $\omega_n(x)$ is the Chebyshev Polynomial of the first kind, that is $$\omega_{n+1}(x)=2^{-n}\cos((n+1)\cos^{-1}(x)$$

I have seen (and proven) that these polynomials are orthogonal w.r.t the weight function $1/\sqrt{1-x^2}$, but I am unable to find any result using the weight function above. When performing the suggested substitution, I end up having to integrate $$\int_0^\pi \cos(j\theta)\cos(k\theta)\sin^2(\theta)d\theta$$ How can I go about integrating this, is there some trig identity I'm not seeing? The product to sum identity doesn't seem to get me anywhere

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    $\begingroup$ the exercise is wrong, the weight is $1/\sqrt{1-x^2}$. The second kind chebyshev polynomials satisfies the exercise orthogonality. $\endgroup$
    – Yimin
    Commented Mar 20 at 1:30
  • $\begingroup$ @Yimin I see, that is unfortunate. Thank you! $\endgroup$
    – modz
    Commented Mar 20 at 1:43

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