\[\large \begin{cases} \tan^{a_{i}} (45^\circ - i^\circ) = \tan (44^\circ - i^\circ) \\ \tan^{b_{i}} (45^\circ + i^\circ) = \tan (46^\circ + i^\circ) \end{cases} \]

Given the above, where \(i=1,2,...,43\), find \[\prod_{i=1}^{43} \dfrac{a_{i}}{b_{i}}\].

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