Reference request: tangent half-angle formula

Reference request: tangent half-angle formula

I keep involuntarily deriving new tangent half-angle formulas. Some of
them will be in a paper I wrote that was recently accepted by the Monthly.
This one I wonder why I never noticed until less than a half-hour ago. The
proof of this is trivial: any mathematician can do it in a minute. But I'm
not sure the result itself is trivial (ask me that two years hence).
My question is: Is this out there somewhere in the literature?
$$\prod_{i=1}^n \tan\frac{\alpha_i}{2} = \frac{\prod_{i=1}^n
\sin\alpha_i}{e_0(\cos)+e_1(\cos)+e_2(\cos)+e_3(\cos)+\cdots}$$
where $e_k(\cos)$ is the $k$th-degree elementary symmetric polynomial in
$\cos\alpha_i,\ i=1,\ldots,n$.