Summary: We offer new proof of the recent sharp trigonometric inequality \(\cos x/\cos(x/2)\geqslant1-4x^2/\pi^2\) for \(x\in(0,\pi/2)\), discovered by J. D’Aurizio [Math. Inequal. Appl. 17, No. 4, 1487–1498 (2014; Zbl 1304.26014)]. The converse inequality, as well as sharp analogous inequalities are pointed out, too.