Let $e_k$ be the $k$th-degree elementary symmetric polynomial in $\tan\theta_1,\tan\theta_2,\tan\theta_3,\ldots$ (and if the sequence of $\theta$s is finite remember that the $k$th-degree elementary symmetric polynomial in
$n<k$ variables is $0$). Then
(e_0-e_2+e_4-\cdots)^2 + (e_1-e_3+e_5-\cdots)^2 = \sec^2\theta_1\sec^2\theta_2\sec^2\theta_3\cdots
I haven't seen this listed among Pythagorean trigonometric identities anywhere, but maybe that means I haven't seen the sources where they would be.
So my question is: Is this "known" in the sense of being found in any authoritative or other published source?