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

Step 4 of 6: Expand the product to order x²

In plain words

For an ordinary finite polynomial, there is a shortcut (Vieta's relations) that reads the sum of the reciprocal roots straight off the coefficient of $x$ : no need to know the roots individually. Euler applied that very same shortcut to his infinite product, formally multiplying it out and reading off the coefficient of $x^2$ .

\prod_{n=1}^{\infty}\left(1-\frac{x^2}{n^2\pi^2}\right)=1-x^2\sum_{n=1}^{\infty}\frac{1}{n^2\pi^2}+O(x^4)

Detailed analysis

Multiplying out infinitely many factors formally, the coefficient of $x^2$ is minus the sum of the reciprocals of the "squared roots" — exactly the rule (Vieta's relations) that gives the sum of reciprocal roots of a finite polynomial from its coefficients.

Terms in this step

Vieta's relations: Formulas expressing the coefficients of a polynomial as sums and products of its roots; in particular the coefficient of $x$ in a polynomial with leading coefficient $1$ and constant term $1$ is minus the sum of the reciprocal roots.

Knowledge used in this step

Power series and Taylor series