# The natural algorithmic approach of mixed trigonometric-polynomial problems

## TL;DR

This paper introduces a new algorithm that simplifies mixed trigonometric-polynomial inequalities to polynomial inequalities, enabling automated proofs and applications in approximations and inequality improvements.

## Contribution

The paper presents a novel algorithm for proving mixed trigonometric-polynomial inequalities by reduction to polynomial inequalities, enhancing automated proof capabilities.

## Key findings

- Successfully applied to rational approximations of cos^2(x)
- Improved a class of inequalities by Z.-H. Yang
- Algorithm suitable for automated proof systems

## Abstract

The aim of this paper is to present a new algorithm for proving mixed trigonometric-polynomial inequalities by reducing to polynomial inequalities. Finally, we show the great applicability of this algorithm and as examples, we use it to analyze some new rational (Pade) approximations of the function $\cos^2(x)$, and to improve a class of inequalities by Z.-H. Yang. The results of our analysis could be implemented by means of an automated proof assistant, so our work is a contribution to the library of automatic support tools for proving various analytic inequalities.