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What is the origin of the formula $$ \prod_{k=1}^n \sin\theta_k= \begin{cases} \displaystyle \frac{(-1)^{\lfloor\frac {n}{2}\rfloor}}{2^n}\sum_{e\in S}\cos(e_1\theta_1+\cdots+e_n\theta_n)\prod_{j=1}^n e_j &\text{if $n$ is even}, \\ \displaystyle \frac{(-1)^{\lfloor\frac {n}{2}\rfloor}}{2^n}\sum_{e\in S}\sin(e_1\theta_1+\cdots+e_n\theta_n)\prod_{j=1}^n e_j &\text{if $n$ is odd} \end{cases} $$ where $e_1, \ldots, e_n\in \{-1, 1\}$ which can be found in the Wikipedia list of trigonometric identities?

Further questions

  • As the related Wikipedia entry doesn't contain noting on these matters, could someone provide me a proof of this result?
  • Moreover is it applicable for real computations?
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    $\begingroup$ The proof uses the first four Product-to-sum identities and induction on $n$. $\endgroup$
    – Somos
    Commented Jun 7, 2023 at 16:34

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