In the identity $ \cos \left( \sum_i \theta_i \right) = \sum_{\text{even }n\ge0} (-1)^{n/2} \sum_{|I|=n} \prod_{i\in I} \sin\theta_i \prod_{i\not\in I}\cos\theta_i $ one can prove the case of finitely many values of $i$ by induction on the number of such values, and questions of convergence are easy to treat when there are infinitely many (and similarly with sines).
In the identity $ \tan \left( \sum_i \theta_i \right) = \frac{e_1-e_3+e_5-\cdots}{e_0 - e_2 + e_4 -\cdots} $ where $e_k$ is the $k$th-degree elementary symmetric polynomial in the variables $\tan\theta_i$, the finite case is similarly routine.
What is known about convergence in the infinite case?
LATER EDIT:
I derived this odd identity that I have not seen elsewhere (so attribute it to me if you mention it in a publication, unless you find it in something earlier):
$ \csc\left( \sum_{i=1}^n \theta_i \right) = \frac{(-1)^{\lfloor(n-1)/2\rfloor}(\csc\theta_1\cdots\cdots\csc\theta_n)}{f_{(n\operatorname{mod} 2)} - f_{(n\operatorname{mod} 2)+2} + f_{(n\operatorname{mod} 2)+4} - \cdots\cdots} $ where
- $f_k$ is the $k$th-degree elemenary symmetric polynomial in the variables $\cot\theta_i$
- $\lfloor a\rfloor$ is the greatest integer $\le a$
- $(n\operatorname{mod} 2)$ is the remainder on division of $n$ by $2$
so that the $\pm$ in the numerator is $ \begin{cases} + & \text{if $n=1$ or $2$} \\ - & \text{if $n=3$ or $4$} \\ + & \text{if $n=5$ or $6$} \\ - & \text{if $n=7$ or $8$} \\ & \text{etc.} \end{cases} $ A funny thing about this is that to get the case $n-1$ from the case $n$, you would presumably just set $\theta_n=0$, but then the cosecant and the cotangent both blow up. So you apply L'Hopital's rule, and fully half of the terms in the denominator vanish, if viewed as terms within $f_k$.
Can anything sensible be said about $\csc\left(\sum_{i=1}^\infty \theta_i \right)$?
And (also my own) $ \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}}. $ so we have $\text{even}\leftrightarrow\text{odd}$ alternation between the numerator and the denominator every time $n$ is incremented by $1$. Similar remarks about L'Hopital apply, and the same question about infinite sums can be asked.