In another posting I wrote about a trigonometric relation I had derived, but that ended up not being the main point of the posting:

Strange pattern in rounding errors?

So as long as we're here, let's make it the main point of *this* posting. I posted something like this a couple of days ago on stackexchange, with no answers yet.

Let's make this *two* questions:

- Is this "known" in the sense of being in some book, refereed paper, or the like?;
- Is there a
*straightforward*way to prove this?

Here's how I derived this relation: I showed that a certain function satisfies a certain differential equation; then I showed that a certain function emerges as the antiderivative that you get by the usual second-semester-calculus methods; then I said these *ought* to be the same thing because they both solve a geometry problem that arose in some amateur cartography of a (maybe?) somewhat impractical sort, therefore they must be the same; then I checked it numerically and it checked. **But there ought to be a more straightforward way.**

Here's the result: If $$\tan\gamma=\dfrac{\sin\alpha\sin\beta}{\cos\alpha+\cos\beta}$$ and $\alpha, \beta, \gamma\in(0,\pi)$ or $\alpha,\beta,\gamma\in(-\pi,0)$ then $$ \tan\dfrac\gamma2=\tan\dfrac\alpha2\cdot\tan\dfrac\beta2. $$

A tangent half-angle formula that everyone knows, or at least that's out there in trigonometry-for-adults books that were occasionally published before about 1930, says
$$
\frac{\sin\alpha+\sin\beta}{\cos\alpha+\cos\beta} = \tan\frac{\alpha+\beta}{2}.
$$
Does it make any sort of sense to say that the fact that what I derived, and this "known" identity are reminiscent of each other has some reason behind it? (So I guess this is really *three* questions.)

proof, this proposition might be considered "trivial". But I think maybe in some of theconsequencesit might not be. But I don't feel like being specific about that just yet. $\endgroup$