Worked solution: Euler's proof via the sine product formula (1734–1735)

The Basel problem asks for the closed form of $\zeta(2)=\sum_{n=1}^{\infty}\frac{1}{n^2}$, posed by Pietro Mengoli in 1644 and unsuccessfully attacked by the Bernoullis and Leibniz. Convergence was never in doubt — the series is dominated by $\int_1^{\infty}dx/x^2$, and by Euler's day roughly six correct digits were known — but an exact expression resisted every approach.

Euler's strategy, recounted step by step in Dunham's account (Ch. 3 of the source), is to obtain two different algebraic descriptions of the same function $\frac{\sin x}{x}$: its Maclaurin series (Step 2) and an infinite product built from its zeros (Step 3). Once both expansions are on the table, equating them coefficient by coefficient converts information about the zeros of $\sin x$ — pure geometry — into arithmetic information about $\zeta(2)$.

The choice of $\sin x$ is the creative act: it is the simplest entire function whose zeros form an explicit, evenly spaced, doubly infinite family $0,\pm\pi,\pm2\pi,\dots$, exactly the kind of data needed for a product whose $x^2$-coefficient is a series of reciprocal squares.