So the proof of $ \sin^2 x + \cos^2 x = 1 $ rests upon Pythagoras' theorem and the definitions $ \sin x := O/H $ and $ \cos x := A/H $, where $O$,$H$ and $A$ are the opposite, hypotenuse and adjacent side of a right angled triangle respectively. However if I was on a general manifold with a metric tensor $g$, then Pythagoras' theorem would read
$$ g_{ij}x^i x^j = c^2 $$
which would not give me the desired equation above. So my question: Does $ \sin^2 x + \cos^2 x = 1 $ only hold in Euclidean space?