This will be a variation on the theme of this question, or maybe a rephrasing of it with a somewhat readjusted emphasis.

In this posting I introduced the function \begin{align} & f_3(\theta_1,\theta_2,\theta_3,\ldots) \\[8pt] = {} & \sum_{\text{odd } n\,\ge\,3} (-1)^{(n-3)/2} (n-1) \sum_{|A|\,=\,n} \prod_{i\,\in\,A} \sin\theta_i\sum_{i\,\notin\,A} \cos\theta_i \tag{$f_3$} \end{align} and the conditional trigonometric identity $$ \text{If } \sum_i\theta_i=\text{half-circle then } f_3(\theta_1,\theta_2,\theta_3,\ldots) = \frac 1 2 \sum_i \sin(2\theta_i). \tag{area} $$ (In $(f_3),$ each term has finitely many sine factors and cofinitely many cosine factors – a bit of terminological synchronicity.)

In a polygon inscribed in a circle, at one of the vertices, let $\theta_1$ be the angle between the tangent line to the circle and the adjacent side, and $\theta_2$ the angle between that side and the next diagonal, and $\theta_3$ the angle between that diagonal and the next, and continue that way until you get the angle between the next side and the opposite tangent ray. At the next vertex the whole sequence $\theta_1,\theta_2,\theta_3,\ldots$ will get cyclically shifted by one place. If the circle has unit diameter (not unit radius) then the right side of the line labeled $(\text{area})$ above is the polygon's area, so therefore so is the left side.

In that same posting I introduced the function \begin{align} & f_2(\theta_1,\theta_2,\theta_3,\ldots) \\[8pt] = {} & \sum_{\text{even } n\,\ge\,2} (-1)^{(n-2)/2} n \sum_{|A|\,=\,n} \prod_{i\,\in\,A} \sin\theta_i\sum_{i\,\notin\,A} \cos\theta_i \tag{$f_2$} \end{align} and also this identity: $$ \text{If } \sum_i\theta_i=\text{half-circle then } f_2(\theta_1,\theta_2,\theta_3,\ldots) = \sum_i \sin^2\theta_i. \tag{what?} $$

Here I knew no geometric interpretation like that for $f_3.$

I left this aside for a long time, and then a few days ago it occured to me that

• As the maximum distance between vertices of that cyclic polygon approaches $0,$ then of course so does the expression in the line labeled $(\text{what?});$ and

• As the maximum distance between vertices of that cyclic polygon approaches $0,$ then so does the difference between the areas of that inscribed polygon and the circumscribed polygon that touches the circle at the vertices of the inscribed polygon; and

• That difference in areas should be some polynomial function of the sines and cosines of the $\theta\text{s}.$

So it occurred to me that that difference in areas might be the geometric interpretation of the identity $(\text{what?})$ that had eluded me.

But it is a straightforward exercise to show that that is false. The value of $(\text{what?})$ is bigger that that area.

But now I wonder whether the conjunction of my bullet points above might suggest anything about the geometry of $(\text{what?}).$

(I also posted another question here about my nlab posting.)