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Geometry Level 4

\[\sin x+\tan^2 x+\sin^3 x+\tan^4 x+\cdots+\sin^{2n-1}x+\tan^{2n}x>\sqrt{2}\] With \(n\in\mathbb{N^*}\) and \(x\in \left(0;\frac{\pi}{6}\right)\), what can be said about the above inequality.

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