Question: Show that $$\dfrac{x^2 + x^{-2}}{x-x^{-1}} \geq 2 \sqrt{2}$$ for $x > 1$.
My attempts: After spending some time trying to prove it by $AM-GM$ and with algebraic manipulation, I tried to use trigonometric substitutions like letting $x = \tan\theta$ and $x = \sin\theta$ although I was still unsuccessful. I know that this can be proven with calculus, however I am looking to prove this without the aid of calculus. Any help would be appreciated!