Long way, long day

The $LHS$ can be written as $LHS=\bigg[sin^6\big(\frac{x}{2}\big)+cos^6\big(\frac{x}{2}\big)\bigg]+\bigg[\frac{1}{sin^6\big(\frac{x}{2}\big)}+\frac{1}{cos^6\big(\frac{x}{2}\big)}\bigg]+4$ $=\bigg[sin^2\big(\frac{x}{2}\big)+cos^2\big(\frac{x}{2}\big)\bigg]^2-3sin^2\big(\frac{x}{2}\big)cos^2\big(\frac{x}{2}\big)+\frac{1-3sin^2\big(\frac{x}{2}\big)cos^2\big(\frac{x}{2}\big)}{sin^6\big(\frac{x}{2}\big)cos^6\big(\frac{x}{2}\big)}+4$ $=5-\frac{3}{4}sin^2(x)+\frac{64-48sin^2(x)}{sin^6(x)}\geq 5-\frac{3}{4}+64-48=\frac{81}{4}\geq LHS$ So the equality happens when $\begin{cases} sin^2(x)=1\\sin^6(x)=1\\cos^2(4x)=1\end{cases}$ , solve the system and we get $x=\frac{\pi}{2}+k\pi; k\in\mathbb{Z}$ . Due to the interval, we'll have $\frac{\pi}{2},\frac{3\pi}{2},\frac{5\pi}{2},\frac{7\pi}{2}$ as roots and the sum of them $\approx 25.13$