To find the Taylor polynomial of a composition of functions, such as e^{\cos x}, one can start by expressing each function as its own Taylor series. For instance, near π/2, e^{\cos x} can be expanded using the Taylor series for cos x, which itself can be approximated around that point. The challenge lies in composing these series, and calculating derivatives may simplify the process depending on the desired outcome. A useful method for determining coefficients and constructing the entire series is outlined in the Taylor series documentation. This approach provides a systematic way to derive the polynomial for complex function compositions.