Proof of a trigonometric inequality in $(0,\pi/2)$ I want to show
$$f(x)=-512\sin\frac{4x}7+1048\sin\frac{6x}7-800\sin\frac{8x}7+216\sin\frac{10x}7>0\quad x\in(0,\pi/2)$$
This trigonometric inequality has been verified by Mathematica using the Plot commend. However, I cannot give a rigorous proof of it. Any suggestion, idea, or comment is welcome, thanks!

Let $\cos\frac{2x}{7}=t$.
Hence, $$f'(x)=\frac{16}{7}(t-1)^2(270t^3+140t^2-131t-34)$$
By the way, $t=\cos\frac{2x}{7}>\cos\frac{\pi}{7},$ which says that $f'(x)>0$
and $f(x)>f(0)=0$.
