Reduced Chebyshev approximation?

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Few days ago my teacher mentioned a method of approximation for a function. I think it was called "reduced chebyshev approximation", where you find the taylor series of degree $N$ then subtract from it the chebychev polynomials of degree $N-1$. Does anyone know what my teacher was talking about? I have been looking online for days to find this kind of approximation but couldn't find any.