On generalization of D'Aurizio-S\'andor trigonometric inequalities with a parameter

In this work, we generalize the D'Aurizio-S\'andor inequalities (\cite{D'Aurizio,Sandor}) using an elementary approach. In particular, our approach provides an alternative proof of the D'Aurizio-S\'andor inequalities. Moreover, as an immediate consequence of the generalized D'Aurizio-S\'andor inequalities, we establish the D'Aurizio-S\'andor-type inequalities for hyperbolic functions.