The article initially points out that in mathematics, there are often equivalent definitions, that each have their own benefits and drawbacks. I think the autho

The article initially points out that in mathematics, there are often equivalent definitions, that each have their own benefits and drawbacks. I think the author could have just written "I have found an alternative approach" instead of "I have found a better approach". As others have noted here, the geometric argument only works "intuitively" for acute angles and the functions have to be explicitly extended. Still, I hadn't seen this proof of the angle addition formula yet, and I found it neat. From a point of view of formalisation, a power series based approach (either directly or via the complex exp function) as traditionally used is probably better, because going into analysis (especially complex analysis), you're going to need power series anyway. Geometry meanwhile is intuitive to us but you'd have to encode a bunch of Euclidean axioms and theorems beforehand, which you might not otherwise use. Also, for better or worse, many mathematicians aren't really taught axiomatic geometry (I wasn't at least).