This is a weirdly elementary question related to the history of real-valued Fourier expansions for which I cannot find any good references. For simplicity, I am following Wikipedia's conventions with regards to notation: https://en.wikipedia.org/wiki/Fourier_series and https://en.wikipedia.org/wiki/Spherical_harmonics
Consider the truncated expansion of a suitable (square-integrable) function $f(\theta,\phi)$ on $S^2$ (sphere) in the real spherical harmonics: $$f(\theta, \varphi) = \sum_{\ell=0}^N \sum_{m=-\ell}^\ell f_{\ell m} \, Y_{\ell m}(\theta, \varphi)$$ Similarly, consider the truncated Fourier expansion on the real line of a suitable function, using the amplitude-phase form of the basis functions: $$s(x) = \frac{A_0}{2} + \sum_{n=1}^N A_n\cdot \cos\left(\tfrac{2\pi }{P}nx - \varphi_n \right)$$ or, equivalently, the sine-cosine expansion: $$s(x) = \frac{a_0}{2} + \sum_{n=1}^N \left(a_n \cos\left(\tfrac{2\pi}{P} nx \right) + b_n \sin\left(\tfrac{2\pi}{P} nx \right) \right)$$
Did anyone, throughout history, ever try to introduce an alternative function to $cos$ and $sin$ that would make it possible to write the Fourier expansion more similar to the spherical harmonics, along the lines of: $$s(x) = \frac{B_0}{2} + \sum_{n=1}^{2N} B_n\cdot C_n\left(\tfrac{2\pi }{P}nx\right)$$ or even: $$s(x) = \sum_{n=0}^{2N} B_n\cdot C_n\left(\tfrac{2\pi }{P}nx\right)$$ if we define: $$C_0=\frac{B_0}{2}$$ where for $n>0$ the function $C_n(x)$ is simply alternating between $cos$ and $sin$, analogously to how $Y_{lm}(\theta,\varphi)$ is constructed.
Doing this would seem awkward today but I am interested if there was any point in history where serious mathematicians used such definitions and if they would come with any benefits. One potential benefit is when writing program code where this could give slightly more compact code. For the complex case we of course have the use of $exp$ and my suspicion is that although you have to deal with complex values (not a big deal at all) it is "handy-enough" to kill off any use similar to a function $C_n(x)$ . Any comments are appreciated.