As others have said, the proof relies on some trig facts (Though maybe you can replace `sin` with any zero-crossing periodic function). But it's worse than tha

As others have said, the proof relies on some trig facts (Though maybe you can replace `sin` with any zero-crossing periodic function). But it's worse than that. The only proof I can see for the second equality there depends on showing the numerator is composite. And that in turn depends on something very like the argument you mention. So unless there is a much simpler proof of that equality, the proof is only a "one liner" because it leaves Euclid's argument as an exercise to the reader.