Alternative proof $g(u)=6 + 5 \sin u + \sin(2 u)- \cos u - \cos(2 u) \ge 0$ for $u\in\left[-\frac \pi 2, \frac \pi 2\right]$

The given inequality

$$g(u)=6 + 5 \sin u + \sin(2 u)- \cos u - \cos(2 u) \ge 0$$

for $u\in\left[-\frac \pi 2, \frac \pi 2\right]$, comes out from an answer given to this other recent question.

The solution given there makes use of elementary analysis tools, mainly derivatives, and some numerical check.

Can we find an alternative and more effective or elegant proof?