This answer to a related question notes that in addition to the usual Fourier expansion of $\sin^2(x)=\frac12 -\frac{\cos2x}2$
we do have the freedom to extend $\sin^2(x)$ to an odd function on $[−\pi,\pi]$ instead, in which case the Fourier series will contain only sine functions
I didn't know that. What does that look like, even on all $x$ (not just $[−\pi,\pi]$)? I can't seem to find it online anywhere. It should be something like $\Sigma^\infty a_n\sin nx$ but what are the coefficients?
There is an understandable cognitive dissonance created by questions like this: a function whose natural definition makes it continuous with continuous derivatives of all orders, and is periodic (say with period $2\pi$), is used to describe an "artificially/prankishly(?) created" function which is equal to the natural function on some interval, but is then extended (!!!) in some semi-random but not unreasonable way to be (maybe) a periodic function which may fail to be continuous or indefinitely differentiable, etc. Then the exercise of computing the somewhat-artificial functions' Fourier coefficients causes cognitive disturbance in juxtaposition with the natural function.
Perhaps this slightly-meta answer makes @NicholasStull's comments sufficient for the OP to carry out the exercise.