You don't need to know the proofs for Simpson's rule of Trapezoidal rule I think (to be absolutely sure, check the syllabus on the BOSTES site, but they never ask about those). Trapezoidal rule formula is easy to prove though, it's just the area of a trapezium. And what are your referring to by 'log proofs'?
Iirc there are things students are technically expected to be able to prove, e.g. certain circle geometry theorems, but they rarely get asked.
There would be guidance for those types of questions however.
Probably. Everything has guidance nowadays. Anyway, here is the syllabus for Mathematics 2U/3U Year 11-12: http://www.boardofstudies.nsw.edu.au/syllabus_hsc/pdf_doc/maths23u_syl.pdf
When I was taught the circle geometry theorems we did go through how to prove them (except for the radius is perpendicular to tangent drawn to point of contact) however I still have not seen anything ever get asked.
People who don't read the syllabus could be in trouble if the HSC exam ended up asking for a proof of one of the circle geometry theorems. But you're right, it'd be quite surprising if the HSC exam asked for a proof of such a theorem.
Oh, and for the radius is perpendicular to tangent one, that's one that's meant to have its proof discussed in class, but the reproduction of its proof is not needed by students according to the syllabus. So maybe that's why your teacher didn't bother proving it.
It's proof by contradiction if I recall?
Yeah, proof by contradiction is a way to prove that radius one. A proof is here on ProofWiki: https://proofwiki.org/wiki/Radius_at_Right_Angle_to_Tangent
