The background is from a highly cited paper "Improved Approximation Algorithms for Maximum Cut and Satisfiability Problems Using Semidefinite Programming".
I know how to prove $\frac{2\theta}{\pi}\ge \rho(1-\cos \theta)$ for $\theta\in [0, \pi]$, where $\rho=0.87856$.
But I get stuck in the proof of a related inequality $2-\frac{2\theta}{\pi}\ge \rho(1+\cos \theta)$ for $\theta\in [0, \pi]$?