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

Step 5 of 6: Match coefficients of x²

In plain words

Two different recipes for baking the same cake must list the same amount of every ingredient. Here the "cake" is the function $\sin x/x$ , the two "recipes" are the Maclaurin series from Step 2 and the expanded product from Step 4, and matching the $x^2$ ingredient is exactly the numerical identity Euler was chasing.

-\frac{1}{3!}=-\frac{1}{\pi^2}\sum_{n=1}^{\infty}\frac{1}{n^2}\ \Longrightarrow\ \sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}

Detailed analysis

Steps 2 and 4 give two power series for the same function $\sin x/x$ , so their coefficients must agree term by term. Comparing the coefficient of $x^2$ turns the geometric statement about the zeros of $\sin x$ directly into the numeric identity Euler was after.