Here was my previous question: Proof by Induction Using Fibonacci numbers
There was a similar one in existence already over here: Inductive proof of a formula for Fibonacci numbers
The one that was answered already had a decent answer but it isn't as detailed as I wish it could be.
hypergeometric mentioned this in his/her answer:
"Note also that 1+1/ϕ=ϕ and 1−ϕ=−1/ϕ."
However, I don't know where this was obtained. Everything else I can see, but this is not immediately clear to me. Any help would be appreciated.