Kudos for the kids for putting in the effort. However, without seeing any details (and no experts were cited in the article!) I would assume what was done was rather trivial. I suppose if you define the trigonometric functions using power series you could probably arrive to the result rather easily. I didn't work out the details, but that would seem like an alternative way to get a handle to these functions without using any geometric ideas.

Edit: so they refer to a book by a math PhD for the claim that there is no trigonometric proof. The book is rather old, so maybe the information is outdated, but it does suggest it's maybe not entirely trivial.

Edit2: wait, wouldn't this just boil down to showing that sin^2 x + cos^2 x = (isin x + cos x)(-isin x + cos x) = e^(ix)e^(-ix)=1?

Surely you'd have to do a bit of work with the series to be able to write that, but looks like standard calculus.