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Given that 0<α,β<π40<\alpha,\beta<\dfrac {\pi}{4}0<α,β<4π such that cos(α+β)=45 \cos (\alpha + \beta) = \dfrac {4}{5} cos(α+β)=54 and sin(α−β)=513\sin (\alpha - \beta) = \dfrac {5}{13} sin(α−β)=135 holds true, find tan2α\tan 2\alphatan2α.
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