hi
taylor series is a fundamental concept in analysis and the beauty of it is with the help of it we can represent any trigonometric function in terms of algebraic expansions( that is in terms of polynomial expansions or series) .
AS we know that trigonometric functions are transcendental functions taylor series give the expansion of transcendental in to algebraic form.
also with the hep of taylor series we can tell whether a given function is analyic or not with out using any other mathods.