Help proving $\forall x \geq 2: \cos(2 \pi/x) \leq 1 - 1/x^2$ I want to prove the following inequality: 
$$\forall x \geq 2: \cos(2 \pi/x) \leq 1 - 1/x^2  
$$
I believe it holds: numerical evidence.
I tried to prove it using taylor series, but it didn't work so well. Any ideas?

Note that by Taylor's series $\forall x$
$$\cos x\le 1-\frac{x^2}2+\frac{x^4}{24}$$
thus
$$\cos \frac{2\pi}{x}\le 1-\frac{2\pi^2}{x^2}+\frac{16\pi^4}{24x^4}\le 1-\frac{1}{x^2}$$
indeed
$$1-\frac{2\pi^2}{x^2}+\frac{2\pi^4}{3x^4}\le 1-\frac{1}{x^2}\iff\frac{1}{x^2}-\frac{2\pi^2}{x^2}+\frac{2\pi^4}{3x^4}\le0$$
$$\iff 3x^2-6\pi^2x^2+2\pi^4\le0\iff x^2(6\pi^2-3)\ge2\pi^4\iff x^2\ge \frac{2\pi^4}{6\pi^2-3}\approx3.465$$
