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Special values of Chebyshev polynomials

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I need to use the generating function for Chebyshev polynomials to show the special values. I am easily able to show these using T_n(x) = cos(ncos^-1(x)) but cannot figure it out using the generating function. I have tried plugging x = 0, or -1, or 1 into both sides of the generating function and using the geometric series sum formula on the left hand then simplifying to obtain the results but cannot get them. Need a solution with details please.

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This solution shows step-by-step calculations to determine the special values for the Chebyshev polynomials.

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