Oh how great our vision is in hindsight! In a way, yes.. If you’re clever enough to come up with trig sub before it was invented.. or you knew the derivative or arctan(x) The beauty of this trick is that it doesn’t require anything except knowledge of derivatives
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Yep, and x = tan(t) with its derivative is half of the chain rule needed for the inverse theorem. Justification of this is the second half. Once you switched to the "no integrals here, just derivatives" interpretation, you started deriving the inverse theorem in disguise.
You don’t need chain rule for the derivative of tan. Quotient rule does the trick