Long way, long day

Geometry Level 5

\[ \bigg[\cos^3\bigg(\frac{x}{2}\bigg)+\frac{1}{\cos^3\big(\frac{x}{2}\big)}\bigg]^2+\bigg[\sin^3\bigg(\frac{x}{2}\bigg)+\frac{1}{\sin^3\big(\frac{x}{2}\big)}\bigg]^2=\frac{81}{4}\cos^2(4x)\] Find the sum of all real \(x\in [0;4\pi]\) satisfy the equation above, submit your answer to 2 decimal places. If you think there are no roots, enter 0.00.

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