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Evaluate \[\large \dfrac{\displaystyle \int_0^{\frac{\pi}{2}} \! \sin^{2n+2}\left(x\right) \, \mathrm{d}x}{\displaystyle \int_0^{\frac{\pi}{2}} \! \sin^{2n}\left(x\right) \, \mathrm{d}x}\] for all positive integer \(n\).

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