Can I prove Pythagoras' Theorem using that $\sin^2(\theta)+\cos^2(\theta)=1$?

The usual proof of the identity $\cos^2 t+\sin^2 t=1$ uses the Pythagorean Theorem. So a proof of the Pythagorean Theorem by using the identity is not correct.

True, we can define cosine and sine purely "analytically," by power series, or as the solutions of a certain differential equation. Then we can prove $\cos^2 t+\sin^2 t=1$ without any appeal to geometry.

But we still need geometry to link these "analytically" defined functions to sides of right-angled triangles.

Remark: The question is very reasonable. The logical interdependencies between various branches of mathematics are usually not clearly described. This is not necessarily always a bad thing. The underlying ideas of the calculus were for a while quite fuzzy, but calculus was still used effectively to solve problems, Similar remarks can be made about complex numbers.

My take on this is that in Euclidean space the Pythagorean theorem is equivalent to $\sin^2(\theta)+\cos^2(\theta)=1$. One simply uses similar triangles - every right-angled triangle is similar to a triangle with hypotenuse $1$. The sin and cos functions make sense in the Euclidean plane because similarity preserves the ratios between lengths and the angles between lines.

There are some quite deep geometrical ideas here. In non-euclidean geometry we don't have the same simple scale invariance (similarity) to work with. So the parallel postulate is essential to the proof.