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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 $a_n$, $b_n$?
Depends on the domain you're working over.
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 $\Bbb{R}$).
