Quite possibly DeMoivre's theorem and definitely it's application. Possibly some compound angle formulae proofs along with a possibly ''prove root two is irrational'' and probably a proof by induction. I think louisoc is the only madman.
Compound angle formulae? Sounds distinctly trigonometric to me, and, last time I checked, trigonometry was a paper 2 topic.
Proof by contradiction is hardly a theorem. I'll admit, I didn't know we had to prove DeMoivre's until just this morning, but it's really only a standard inductive proof.
Proof of the amortization formula can come up on paper 1, even though it isn't covered in Active Maths.
A full q on financial maths is due this year, so this proof is looking likely.
It's on page 14 of this pdf http://www.projectmaths.ie/documents/modulars/4/FinancialMathsExtraQuestions.pdf
I can quite joyfully say that I was wholly misguided.
In my defence, SEC has been made papers quite unpredictable this year, as seen in the Irish poems and stories and relatively shocking lack of Yeats.
However, it was merely an educated assumption and by no means an assertion.