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.<p>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 The book is rather old, so maybe the information is outdated, but it does suggest it's maybe not entirely trivial.<p>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?<p>Surely you'd have to do a bit of work with the series to be able to write that, but looks like standard calculus.<p>Edit3: ok, so I dug a bit deeper and found that the book that is cited claims as a fact that you cannot prove Pythagorean theorem using trigonometric functions, since to use trig. functions you need Pyth. theorem. This is not true, if you start with series expansions. So I wonder if the story is that the kids relied on the book as a fact, did some calculus and proved their theorem. The teachers, not being mathematicians, took the book as face value as well (and here's where the mistake happened) didn't consult an actual expert before speaking to the press. Still, I'd say pretty nice job from the kids to actively pursue a mathematical problem, hopefully next time with a proper supervisor.