Can You Divide When Formally Proving Trig Identities?

[filter54321]

When formally proving trig identities using algebra are you allowed to divide? Ordinarily I wouldn't think twice about this. Sure, the trig function by which you are dividing might take on 0, but not for all angles. We divide by potential zeros all the time.

My advisor was being very coy about the answer and it was kind of obnoxious. I know that in some constructions of the rational numbers from the integers you can't formally "divide" because, given the integers, you can't define it as an operation.

Is the trig "issue" related? I can't find anything on Google so I think he's being overly Socratic.

 

[mathman]

If it is an identity, then you will end up with a correct formula for all cases when you are not dividing by 0. Those cases can then be inferred as limits.

 

[AlephZero]

When you are "formally proving trig identities", most likely the purpose of the exercise is to make you practise using standard formulas (e.g cos^2 x + sin^2 x = 1, etc) not worrying about the finer points of algebra. You don't normally bother about the fact that functions like tan, cot, sec, cosec are undefined for some angles either.

It would be pedantic to add "except when x = some particular values" to every exercise like this. It wouldn't add any value to the exercise, and might make some students go off in completely the wrong direction.

Of course if you deliberately introduce a new factor like (1 - cos^2 x - sin^2 x) which is identically zero, you will probably get what you deserve (i.e. zero marks!)