On various occasions people have asked here how to prove (but that is NOT the question here) that $$ \text{if } \alpha+\beta+\gamma = \pi \text{ then } \frac{\sin(2\alpha) + \sin(2\beta) + \sin(2\gamma)} 2 = 2\sin\alpha\sin\beta\sin\gamma. $$ I first saw this identity in 2006 when it got added to Wikipedia's List of trigonometric identities by a logged-in user called Sriramoman, who alleged that it occurred perenially on the Joint Entrance Examination of the Indian Institutes of Technology.
It can be proved by methods that we all learned in secondary school, but I prefer a geometric proof that views the two equated quantities as different ways of expressing the area of a triangle inscribed in a circle of unit radius and in which the three angles are $\alpha,\beta,\gamma$. (That is why I wrote it with factors of $1/2$ on the left and $2$ on the right rather than $1$ on the left and $4$ on the right.) (The fact that there is yet another reason to write it that way will wait for another occasion . . . )
My question is whether this identity has known applications, in physics, engineering, graphics, surveying, cartography, mathematics, or anything else?
