In one of Wikipedia's proofs of the solution to the Basel problem, they state that the $x^2$ coefficient of $$\begin{align*} \frac{\sin(x)}{x} &= \left(1 - \frac{x^2}{\pi^2}\right)\left(1 - \frac{x^2}{4\pi^2}\right)\left(1 - \frac{x^2}{9\pi^2}\right)\cdots \end{align*}$$ can be proven inductively to be $$-\left(\frac{1}{\pi^2}+\frac{1}{4\pi^2}+\frac{1}{9\pi^2}\cdots\right)$$

I do not see any way of proving this by induction. I haven't ever come across this use of induction before actually. Could someone point me in the right direction of how to prove this using induction? I don't even know what the inductive hypothesis would be.

I can see this will certainly be the case though as the only way to have $x^2$ terms is by having each $x^2$ term in each bracket only multiplied together by the $1$'s inside the other brackets.

Thanks for your help.