From the identity
![{\displaystyle \sin \left(\sum _{i=1}^{\infty }\theta _{i}\right)=\sum _{{\text{odd }}k\geq 1}(-1)^{(k-1)/2}\sum _{\begin{smallmatrix}A\subseteq \{\,1,2,3,\dots \,\}\\\left|A\right|=k\end{smallmatrix}}\left(\prod _{i\in A}\sin \theta _{i}\prod _{i\not \in A}\cos \theta _{i}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ac33fc1305bf8a5ae4aa995b4038addad7ffe3a)
one can infer that
![{\displaystyle \sin(n\theta )=\sum _{{\text{odd }}k\in \{1,\dots ,n\}}(-1)^{(k-1)/2}{n \choose k}\sin ^{k}\theta \cos ^{n-k}\theta .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55db6b7ab2ac2131524414784babf8d4a1b55db9)
Letting x = nθ, we have
![{\displaystyle {\begin{aligned}\sin x&=\sum _{{\text{odd }}k\in \{1,\dots ,n\}}(-1)^{(k-1)/2}{n \choose k}\sin ^{k}\theta \cos ^{n-k}\theta \\[12pt]&=\sum _{{\text{odd }}k\in \{1,\dots ,n\}}\left[(-1)^{(k-1)/2}\left({n(n-1)\cdots (n-k+1) \over k!}\right)\left({x \over n}\right)^{k}\right.\\&{}\qquad \qquad \qquad \qquad \qquad \qquad \qquad \left.{}\cdot \left({\sin(x/n) \over x/n}\right)^{k}\cos ^{n-k}(x/n)\right]\\[12pt]&=\sum _{{\text{odd }}k\in \{1,\dots ,n\}}\left[(-1)^{(k-1)/2}\ {x^{k} \over k!}\cdot {}\right.\\&{}\qquad \qquad \left.{n(n-1)\cdots (n-k+1) \over n^{k}}\cdot \left({\sin(x/n) \over x/n}\right)^{k}\cos ^{n-k}(x/n)\right]\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9a5b497d3434045acfd0f445a3e87a03e0d4721)
Having reached this point, Euler said that if n is an infinitely large integer, then the three factors on the last line above are equal to 1. In modern language, we would say that as n → ∞, the factors on the last line approach 1 and we get
![{\displaystyle \sin x=\sum _{{\text{odd }}k\geq 1}(-1)^{(k-1)/2}\ {x^{k} \over k!}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7935f621a04fdba307ca69af78ee1e6884084f07)
However, a problem arises concerning the interchange in the order of two limiting operations.
![{\displaystyle \cos x=\sum _{{\text{even }}k\geq 0}(-1)^{k/2}\ {x^{k} \over k!}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9cabdbe7d368c9db61bc45f2960e7ed1d848ee9c)
![{\displaystyle \cos \left(\sum _{i=1}^{\infty }\theta _{i}\right)=\sum _{{\text{even }}k\geq 0}~(-1)^{k/2}~~\sum _{\begin{smallmatrix}A\subseteq \{\,1,2,3,\dots \,\}\\\left|A\right|=k\end{smallmatrix}}\left(\prod _{i\in A}\sin \theta _{i}\prod _{i\not \in A}\cos \theta _{i}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e31af535452dd800f731209364b525a63b490d55)