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I don't know anything about several complex variables nor whether that topic will answer my questions below, but in one complex variable one learns that since $\sin x$ and $\cos x$ are entire functions, their expansions in powers of $x$ have infinite radius of convergence. Since $\tan x$ and $\sec x$ have poles at $\pi/2$, their expansions have radius of convergence $\pi/2$. And since $\cot x$ and $\csc x$ have poles at 0, you can't expand them in non-negative powers of $x$.

"Everybody knows" the two identities $$ \cos\left(\sum_{i = 1}^\infty \theta_i\right) = \sum_{\text{even }n\ge 0} (-1)^{n/2} \sum_{|A| = n} \prod_{i\in A} \sin\theta_i \prod_{i\not \in A}\cos\theta_i $$ $$ \sin\left(\sum_{i = 1}^\infty \theta_i\right) = \sum_{\text{odd }n\ge 1} (-1)^{(n-1)/2} \sum_{|A| = n} \prod_{i\in A} \sin\theta_i \prod_{i\not \in A}\cos\theta_i $$ The following is also found in plenty of books: $$ \tan\left(\sum_{i = 1}^n \theta_i\right) = \frac{e_1 - e_3 + e_5 - e_7 + \cdots}{e_0 - e_2 + e_4 - e_6 + \cdots} $$ where $e_k$ is the $k$th-degree elementary symmetric polynomial in $x_i = \tan\theta_i,\quad i = 1,\dots, n$ and there are only finitely many terms in each sum. The following may possible not appear in print anywhere. I added it to Wikipedia's "list of trigonometric identities" a year or two ago and no one has (yet) stepped in to object that it violates Wikipedia's policy forbidding "original research"): $$ \sec\left(\sum_{i = 1}^n \theta_i\right) = \frac{\sec\theta_2 \cdots \sec\theta_n}{e_0 - e_2 + e_4 - e_6 + \cdots}. $$ The ones that have poles at 0 have more elaborate behaviors: $$ \cot\left(\sum_{i = 1}^n \theta_i\right) = (-1)^{n+1}\left(\frac{f_1 - f_3 + f_5 - \cdots}{f_0 - f_2 + f_4 - \cdots}\right)^{(-1)^{n+1}} $$ where $f_k$ is the $k$th-degree elementary symmetric polynomial in $x_i = \cot \theta_i, \quad i = 1,\dots, n$ (again, all sums are finite). Notice that in the earlier identities, setting any variable to 0 has the same effect as just discarding all expressions involving that variable. But that makes no sense here: it gives us 0/0, and look at how the roles of odd and even trade places when $n$ is incremented or decremented. But L'Hopital's rule resolves everything; you actually get only half as many terms in each of the numerator and the denominator when you drop one variable.

The expression for the other function with a pole at 0 is even more complicated: $$ \csc\left(\sum_{i = 1}^n \theta_i\right) = \frac{(-1)^{\lfloor (n-1)/2 \rfloor} \csc\theta_1 \cdots \csc\theta_n }{ f_{n \mod 2} - f_{n \mod 2 + 2} + f_{n \mod 2 + 4} - \cdots } $$ and again we need L'Hopital's rule when we set a variable to 0.

My question is whether one can make rigorous a parallel between the observations in the first paragraph and the later observations. With the two entire functions, one had sums that always converge and have infinitely many terms. With the two functions with finite radius of convergence one works only with finite sums. With the two functions with poles at 0, one has complications.

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You should be able to deduce everything by writing it all in terms of the complex exponential. Any wonky behavior should in principle be due to something wonky you're doing to the complex exponential. –  Qiaochu Yuan Jun 23 '10 at 22:57
    
Well, that seems to help identify the precise location of any "wonkiness". But the question is still there. –  Michael Hardy Jun 25 '10 at 2:55

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