[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.