I have the inequality $\cos^2(−\frac{\pi}{4}+\frac{\pi}{p})>\frac{1}{2}+\frac{π}{p^2}$​​ and have found that it holds for $p>p^*$, where $p^*$ is some positive number (around ~2.8). I'm looking for a way to determine $p^*$ and to prove the inequality for $p$ greater than this threshold. Any suggestions on approaches or relevant theorems would be greatly appreciated.

Thank you!