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Proof of a Chebyshev Identity

Reference: Wiki on Chebyshev Polynomials, Proof problem 6.

\(\huge T_n(x) = \frac{ ( x - \sqrt{ x^2 - 1} )^n + ( x + \sqrt{ x^2 - 1 } ) ^ n } { 2} .\)


Solution: \(\text{From the above identity, we get}\) \(\large{ T }_{ n }(\cos { \theta )= } \frac { { \left( \cos { \theta -\sqrt { \cos ^{ 2 }{\theta}- 1 } } \right) }^{ n }+{ \left( \cos { \theta +\sqrt { \cos ^{ 2 }{ \theta } -1} } \right) }^{ n } }{ 2 } =\frac { { \left( \cos { \theta -i\sin { \theta } } \right) }^{ n }+{ \left( \cos { \theta +i\sin { \theta } } \right) }^{ n } }{ 2 }\).

\(\text{Using}\) De Moivre's Theorem, \(\text{we get}\) \(\large\frac { { \left( \cos { \theta -i\sin { \theta } } \right) }^{ n }+{ \left( \cos { \theta +i\sin { \theta } } \right) }^{ n } }{ 2 } =\frac { \cos { n\theta -i\sin { \theta +\cos { n\theta +i\sin { n\theta } } } } }{ 2 } =\cos { n\theta }\).

\(\text{This turns out to be the definition of the}\) Chebyshev Polynomial of the first kind.

\(\therefore\text{ Proved}\)

Note by Swapnil Das
5 months, 3 weeks ago

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Nice work. I have a doubt \(\sqrt{1-cos^2(\theta)}=\sin(\theta)\). Can you please explain how you got \(i\sin(\theta)\) and how \(\sqrt{1+cos^2(\theta)}=i\sin(\theta)\). Svatejas Shivakumar · 5 months, 3 weeks ago

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@Svatejas Shivakumar Correct. That's a typo, thanks for pointing out! Swapnil Das · 5 months, 3 weeks ago

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@Svatejas Shivakumar Yup, I don't understand how Swapnil wrote \(\sqrt{1-\cos^2 \theta}\) and \(\sqrt{1+\cos^2 \theta}\) Nihar Mahajan · 5 months, 3 weeks ago

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@Nihar Mahajan Bro, I am Swapnil. Is it fine now? (It was a typo) Swapnil Das · 5 months, 3 weeks ago

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@Swapnil Das Yep, it is correct now. Sorry, Swaplin was a typo, I meant to type Swapnil :P :P :P Nihar Mahajan · 5 months, 3 weeks ago

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