Why do authors claim that Euler gave no proof to his “$\sin(\pi x)= \pi x\prod\limits_{k=1}^{\infty}\left(1-\frac{x^2}{k^2} \right )$” when…

Question

When he proved the relation between $\pi \cot(\pi x)$ and the harmonic series in "Introductio in analysin infinitorum" which states that $$\pi \cot(\pi x)=\sum_{k \to \infty}^{\infty} \frac{1}{x+k}=\frac{1}{x}+\sum_{k=1}^{\infty}\left(\frac{1}{x+n}+\frac{1}{x-n} \right) \text{ for } x\in \mathbb{R} \backslash \mathbb{Z}.$$ It doesn't take a genius to transform this into an infinite product just by knowing the fact that $$\int \pi \cot(\pi x) = \log\left(\sin(\pi x) \right)+C.$$

So my question is, why does every historian/author claim that Euler's first proof of $\displaystyle \zeta (2)=\frac{\pi^2}{6}$ was not rigourous at all because Euler didn't prove his famous infinite product in his lifetime when the proof of the relation between the cotangent and the harmonic series implies directly his infinite product?

EDIT:I'm sorry for doing this but shameless self bump.

I got no answers and once again, I'm sorry.

Answer 1

I am answering the question in the content, the title is asking something about the infinite product.

why does every historian/author claim that Euler's first proof of $\displaystyle \zeta (2)=\frac{\pi^2}{6}$ was not rigorous at all

Euler's first derivation was done by factoring the infinite series for sin, even Euler himself was not satisfied with that method and although that method was correct in finite cases needed to be justified for infinite case, due to that fact later Euler gave alternative (more)rigorous proofs. (rigorous being what was deemed acceptable by other mathematicians of the time.)

After the first proof was deemed not rigorous by the master himself, no other fact/rigorous proof can change the amount of the rigor of that proof. The infinite product was made rigorous by Weierstrass and his treatment of Entire function theory. :

PS: What ever the first proof lacked in rigor it made up in ingenuity.