I had a Precalc Honors student argue yesterday that the equation (tan θ)(cot θ) = 1 is not an identity, because it's not true when θ = 0, because cot 0 is undefined. It really gave me pause. [1/3]
Of course there are reasons for wanting to exclude 1 from the primes, and an account of those reasons should accompany any principled exposition of the theory of primes -- even if a fully satisfying account cannot be given immediately. Similarly here. Hence the pedagogical agony.