Only in school exercises you can solve a differential equation by expanding a sine function into a Taylor series.

In practical physics computations, the solution of differential equations requires numerical methods that do not use the Taylor series of specific functions, even if the theory used for developing the algorithms may use the Taylor series development of arbitrary functions.

For accurate prediction, the simple pendulum equation also requires in practice such numerical methods, which do not rely on the small-angle approximation that enables the use of the Taylor series of the trigonometric functions, for didactic purposes.