In lecture notes on Fourier Analysis, Chebyshev Polynomials denoted by $T_{n}$ was restated as follows:
$$T_{n}(x) = \cos(n\cos^{-1}(x)) = \sum_{0 \leq j \leq\ n/2}(-1)^{j}{n \choose 2j} x^{n-2j}(1-x^{2})^{j}$$
What's the intuition behind Chebyshev Polynomials what makes them significant, what's the initial motivation ?
They are extremely important in polynomial approximation, because they are sort of a model case: in their beloved region $[-1,1],$ they are $\le1,$ and they attain those values $\pm1$ as often as possible for a polynomial of that degree. That's important, because there's a theorem that the error of the optimal approximation of a continuous function by a polynomial will do exactly that. And that's helpful to construct optimal or at least reasonable approximations by polynomials quite often.