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The problem is that the sin and cos functions the proof relies on are embroiled with pi. It's not obvious whether or not that is a confounding issue or not.

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Sin and Cos are typically defined as “Here’s some weird infinite sums” from which they deduce the behaviours. Indeed, pi is typically defined in terms of the first positive zero of the sin function.

Sin and cos are typically defined as the far edge of the right triangle over the hypotenuse, and the near edge over the hypotenuse.

The infinite sums give us ways to calculate their approximations, and show some of their properties as functions.

It's not. Why would it be?

If you assume at the start of your proof that π is rational, it's not clear that you can then still make use of concepts closely related to π. If those concepts depend in any way on π being irrational, then you can't use them to cleanly arrive at the contradiction.

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