Let $c \not= 1$ be a real positive number, and let $n$ be a positive integer.
Prove that $$n^2 \leq \frac{c^n+c^{-n}-2}{c+c^{-1}-2}.$$
My initial thought was to try and induct on $n$, but the negative powers annoyed be badly.
I still think that induction is the way to go, but maybe it is something more clever than I am doing. Although perhaps induction is unhelpful, and I simply need to use inequalities to do this, like AM-GM and Cauchy-Schwarz. But even there I get very stuck. I have a feeling that there is some neat thing I should do to make this a lot simpler.
Any help would be really appreciated. Thanks in advance.
Hint: the RHS is $$\frac{-(1-c^n)(1-c^{-n})}{-(1-c)(1-c^{-1})}$$
To clarify my bonus hint, we may rearrange the AM-HM inequality to state $$(a_1+a_2+\cdots+a_n)(\frac 1a_1+\frac 1a_2+\cdots+\frac 1a_n)\ge n^2$$