# Inequality with even powers of trigonometric functions

For $m>0$,
$0 < n\leqslant m+1$ ($m,n\in \mathbb{Z}$) , and $0 < a < 1$ , prove that $$2^{n}\cdot \left( a^{n}\cos ^{2m}\dfrac {\pi a} {2}+\left( 1-a\right) ^{n}\sin ^{2m}\dfrac {\pi a} {2}\right) \leqslant\cos ^{2m}\dfrac {\pi a} {2}+\sin ^{2m}\dfrac {\pi a} {2}.$$

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