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?



"Known" Pythagorean identity? (reference request)
I don't have enough reputation to comment, but for what it's worth, if we replace $\tan(\theta\_j)$ with $\lambda\_j$ and set $P(x)=\prod (x-\lambda\_j)$, then the left hand side will be $P(i)P(-i)$ by difference of squares and Vieta's formula, and the right hand side $\prod (1+\lambda\_j^2)$. The equality then follows from $(i-\lambda\_j)(-i-\lambda\_j)=1+\lambda\_j^2$. In this case, all that was used was $\tan^2(\theta)+1=\sec^2(\theta)$.


