\begin{align*} \left(\frac{\sin x}{x}\right)^3 &\ge \left(1-\frac{x^2}{6}\right)^3\\&=1-\frac{x^2}{2}+\frac{x^4}{12}-\frac{x^6}{216} \\&\ge 1-\frac{x^2}{2}+\frac{x^4}{12}-\frac{x^4}{24}\\ &=1-\frac{x^2}{2}+\frac{x^4}{24}\\& \ge \cos x. \end{align*}\\
