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This is false in general.
For example, if $a = -2$ and $b = -1$, we have that $$(-2, 0) = \left(-2, \frac{-2}{\pi}\tan^{-1}{(0)}\right)\subseteq\bigcup_{n\in\mathbb Z}\left(-2, \frac{-2}{\pi}\tan^{-1}(n)\right),$$ which makes the equality impossible.