Is it possible for cosine functions to have Fourier sine series expressions or sine functions to have Fourier cosine series expressions?

Is it possible for cosine functions to have Fourier sine series expressions or sine functions to have Fourier cosine series expressions?

For example, do $\sum\limits_{n=1}^\infty a_n\sin nu=\cos u$ and $\sum\limits_{n=0}^\infty b_n\cos nu=\sin u$ admit solutions for $a_n$, $b_n$?

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On $[0, \pi]$, every continuously differentiable function has both a Fourier sine series and a Fourier cosine series. Since $\cos$ and $\sin$ are both $C^1$, they have a sine series and a cosine series respectively.
On $[-\pi, \pi]$, only even functions have cosine series, and only odd functions have sine series. So $\cos u$ does not have a sine series, and $\sin u$ does not have a cosine series. The series you obtained on $[0,\pi]$ will converge to $\sin |u|$ and $\frac{|u|}{u}\cos u$ respectively on this domain (and hence to the $2\pi$-periodic extensions of these functions on all of $\Bbb{R}$).