[q-big]

> The simplicity of the Taylor series of sine and cosine is irrelevant, there are no important applications for those series.

The addition theorems for trigonometric functions can easily be shown by the multiplication theorem for Taylor series (and adding two Taylor series). This proof would be more convoluted if the Taylor series were not so easy.

Also, because of the simplicity of their Taylor series, one immediately sees that sin and cos are solutions of the ODE y'' = -y.

Another application of the Taylor series is that by their mere existence, sin and cos (as real functions) have a holomorphic extension.

[adrian_b]

The proof of any property of the trigonometric functions is trivial when the sine and the cosine are defined as the odd and even parts of the exponential function of an imaginary argument, and the proof uses the properties of exponentiation.

Any proof that uses the expansion in the Taylor series is a serious overkill.

Moreover, those proofs become even a little simpler when the right angle is used as the angle unit, instead of the radian.

In this case, the sine and the cosine can be defined as the odd and even parts of the function i ^ x.

[ncmncm]

Those are useful symbolically, but not computationally.