Math Prove or Proof

What's the difference between math prove and math proof?

Sample A

Prove the trig identity. Why not proof the trig identity?

Sample B

Proof inequality by induction. Why not prove inequality by induction?

I am not talking about working out math problems here. I need to know when to use prove and when to use proof.

 

prove: means demonstrate the truth or existence of (something) by evidence or argument

example: "you will be asked to prove your identity" (means you will need your photo ID)

proof: evidence or argument establishing or helping to establish a fact or the truth of a statement.
"you will be asked to give proof of your identity" (means you will need to give photo ID)

example in math
Theorem 3.1.
Prove √3 is irrational

Proof.
Suppose not; i.e., suppose √3 element Q.
Then some value of m, n element Z with m and n relatively prime and √3 =m/n.
Then 3 = m^2/n^2, or 3n^2 = m^2.

Thus m^2 is divisible by 3 so by Theorem 2.1, m is also. By definition, m = 3k for some k element Z. Hence m^2 = 9k^2 = 3n^2 and so 3k^2 = n^2.
Thus n^2 is divisible by 3 and again by Theorem 2.1, n is also divisible by 3.
But m, n are relatively prime, a contradiction.
Thus √3 not element Q.

 

Prove lies in the question. The actual work to show what must be proven is the proof.