In the rant I wrote at

http://ncatlab.org/nlab/show/trigonometric+identities+and+the+irrationality+of+pi

I asked: Are these four identities the first four terms in a sequence that continues?

This referred to the identities in the last bullet point above that question.

While we're at it, is there any intuitive geometric interpretation of the identity involving $f_2$?

OK, here are the functions involved:

$$ \begin{align} f_0(\theta_1,\theta_2,\theta_3,\dots) & = \sum_{\text{even }n \ge 0} (-1)^{n/2} \sum_{|A| = n} \prod_{i\in A} \sin\theta_i\prod_{i\not\in A}\cos\theta_i \\ f_1(\theta_1,\theta_2,\theta_3,\dots) & = \sum_{\text{odd }n \ge 1} (-1)^{(n-1)/2} \sum_{|A| = n} \prod_{i\in A} \sin\theta_i\prod_{i\not\in A}\cos\theta_i \\ f_2(\theta_1,\theta_2,\theta_3,\dots) & = \sum_{\text{even }n \ge 2} (-1)^{(n-2)/2} n \sum_{|A| = n} \prod_{i\in A} \sin\theta_i\prod_{i\not\in A}\cos\theta_i \\ f_3(\theta_1,\theta_2,\theta_3,\dots) & = \sum_{\text{odd }n \ge 3} (-1)^{(n-3)/2} (n-1) \sum_{|A| = n} \prod_{i\in A} \sin\theta_i\prod_{i\not\in A}\cos\theta_i \\ f_4(\theta_1,\theta_2,\theta_3,\dots) & = \sum_{\text{even }n \ge 4} (-1)^{(n-4)/2} n(n-2) \sum_{|A| = n} \prod_{i\in A} \sin\theta_i\prod_{i\not\in A}\cos\theta_i \\ f_5(\theta_1,\theta_2,\theta_3,\dots) & = \sum_{\text{odd }n \ge 5} (-1)^{(n-5)/2} (n-1)(n-3) \sum_{|A| = n} \prod_{i\in A} \sin\theta_i\prod_{i\not\in A}\cos\theta_i \\ f_6(\theta_1,\theta_2,\theta_3,\dots) & = \sum_{\text{even }n \ge 6} (-1)^{(n-6)/2} n(n-2)(n-4) \sum_{|A| = n} \prod_{i\in A} \sin\theta_i\prod_{i\not\in A}\cos\theta_i \\ f_7(\theta_1,\theta_2,\theta_3,\dots) & = \sum_{\text{odd }n \ge 7} (-1)^{(n-7)/2} (n-1)(n-3)(n-5) \sum_{|A| = n} \prod_{i\in A} \sin\theta_i\prod_{i\not\in A}\cos\theta_i \\ & \vdots \end{align} $$ In each function the coefficient kills off the terms involving values of $n$ smaller than the index, so that for example we could have said "$\text{odd }n \ge 1$" instead of $\text{odd }n \ge 7$ and it would be the same thing.

Now some facts:

Each $f_k$ is a symmetric function of $\theta_1,\theta_2,\theta_3,\dots$.

0 is an identity element for each of these functions, in the sense that $$f_k(0,\theta_2,\theta_3,\dots) = f_k(\theta_2,\theta_3,\dots).$$

$f_k(\theta_1,\theta_2,\theta_3,\dots) - f_k(\theta_1+\theta_2,\theta_3,\dots) = \left. \begin{cases} k & \text{if }k \ge 2\text{ is even} \\ k-1 & \text{if }k \ge 2\text{ is odd} \end{cases} \right\}\cdot \sin\theta_1\sin\theta_2 f_{k-2}(\theta_3,\theta_4,\dots)$ and $=0$ if $k = 0\text{ or }1$.

Now the sequence of identies: $$ \begin{align} f_0 & = \cos(\theta_1 + \theta_2 + \theta_3 + \cdots) \\ f_1 & = \sin(\theta_1 + \theta_2 + \theta_3 + \cdots) \\ \text{If } \sum_{i=1}^\infty \theta_i = \pi,\text{ then } f_2 & = \sum_{i=1}^\infty \sin^2\theta_i=\frac{1}{2} \sum_{i=1}^\infty (\cos(2\theta_i)-1)\\ \text{If } \sum_{i=1}^\infty \theta_i = \pi,\text{ then } f_3 & = \frac{1}{2} \sum_{i=1}^\infty \sin(2\theta_i) \end{align} $$ The QUESTION is whether these are the first four identities in a sequence that continues beyond this point.

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