No matter where I search, every time if there's an article about Fourier series derivation, the first step made by author is to present the following formula:
$$f(x) = \frac{a_0}{2}+\sum_{n=1}^\infty \left(a_n \cos(\frac{2\pi}{T}nx) + b_n \sin(\frac{2\pi}{T}nx)\right)$$
Then, given that formula, the coefficients $a_n$ and $b_n$ are derived. My question is, how did Fourier come up with this main equation? Was it just a lucky guess? It could have been, because there are various proofs showing that the formula is true.
Here the main equation just appears there without justification, just like formulas for even and odd cases below. It's really hard to call the article a derivation in my opinion.
You will agree that the functions $$x\mapsto\cos(\lambda x),\qquad \lambda>0,$$ and their translates are the "purest", "simplest", "most beautiful", etc., periodic functions you can think of. They are the solutions of the "super simple" ODE $$y''+\lambda^2\>y=0$$ for given $\lambda>0$.
Among all these functions solely the functions $$x\mapsto\cos\left({2n\pi \over T}x\right)$$ and their translates have the prescribed period $T$. It follows that the most general functions that can be built linearly from such pure periodic functions of period $T$ are the functions specified in your question.
Herewith the following grand problem arises: Can every $T$-periodic function $g$ be written in this form, or will after the best conceivable approximation of $g$ by series of this type remain some "nonharmonic remainder"? That the latter is not the case is the essential content of the theory of Fourier series.