TITLE: A "known" Pythagorean identity in algebra?
QUESTION [6 upvotes]: 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?)

REPLY [10 votes]: These are both simple corollaries of
$$\sum_{k\geq 0} t^ke_k(x_1,x_2,\dots)=\prod_{k\geq 0}(1+tx_k).$$
There is a typo in both your identities. They should read,
$$(e_0+e_2+\cdots)^2-(e_1+e_3+\cdots)^2=(e_0+e_1+e_2+\cdots)(e_0-e_1+e_2-\cdots)$$
$$=\prod_{k\geq 0}(1+x_k)\prod_{k\geq 0}(1-x_k)=(1-x_1^2)(1-x_2^2)\cdots,$$
and
$$(e_0-e_2+e_4-\cdots)^2+(e_1-e_3+e_5-\cdots)^2=(\mathfrak {Re}[\prod_{k\geq 0}(1+ix_k)])^2+(\mathfrak{Im}[\prod_{k\geq 0}(1+ix_k)])^2$$
$$=(1+x_1^2)(1+x_2^2)\cdots$$