Some will recognize this as similar to a question I asked before, but I want to ask it without the trigonometry.

Let $e_k$ be the $k$th-degree elementary symmetric polynomial in $x_1,x_2,x_3,\ldots$. If $k$ is more than the number of $x$s, then $e_k$ is the sum of no terms and is $0$. From one POV, the following Pythagorean identities are as elementary as anything not in the high-school curriculum: $$ (e_0+e_2+e_4+\cdots)^2 - (e_1+e_3+e_5+\cdots)^2 = (1-x_1^2)(1-x_2^2)(1-x_3^2)\cdots $$ $$ (e_0-e_2+e_4-\cdots)^2 + (e_1-e_3+e_5-\cdots)^2 = (1+x_1^2)(1+x_2^2)(1+x_3^2)\cdots $$ So are these "known" in the sense of being in refereed publications one could cite? And if not, are they "known" in the sense that some people see them mentioned or explicitly used from time to time?

(And if there should happen to be infinitely many $x$s, could this still be considered only algebra by thinking of these as a sort of "formal" series?)