Infinite products $\prod_{n=1}^\infty\sin\frac1n$ and $\prod_{n=1}^\infty\cos\frac1n$

Question

This is my first question.

What are the values of the infinite products $\prod_{n=1}^\infty\cos\frac1n$ and $\prod_{n=1}^\infty\sin\frac1n$?

Answer 1

For the sines, it is quite simple $$\prod_{n=1}^\infty\sin \left(\frac{1}{n}\right)=0$$ For the cosines the result is just a number $$\prod_{n=1}^\infty\cos \left(\frac{1}{n}\right)\sim0.388536153335175859184329576$$ which is not recognized by inverse symbolic calculators.

However, we can approximate it using logarithms since $$\log \left(\cos \left(\frac{1}{n}\right)\right)=\sum_{p=1 }^\infty (-1)^p\frac{ 2^{2 p-3} (E_{2 p-1}(1)-E_{2 p-1}(0)) }{p (2 p-1)!\,n^{2 p}}$$ which makes $$\sum_{n=1 }^\infty \log \left(\cos \left(\frac{1}{n}\right)\right)=\sum_{p=1 }^\infty \frac{(-1)^{p+1} 2^{2 p-3} (E_{2 p-1}(0)-E_{2 p-1}(1)) \zeta (2 p)}{p (2 p-1)!}$$ which converges quite fast $$\left( \begin{array}{cc} p & \text{result} \\ 10 & 0.388538882923887752891524656369 \\ 20 & 0.388536153510144960460336253991 \\ 30 & 0.388536153335190159575729146913 \\ 40 & 0.388536153335175860483365339921 \\ 50 & 0.388536153335175859184454814359 \\ 60 & 0.388536153335175859184329588232 \\ 70 & 0.388536153335175859184329575688 \\ 80 & 0.388536153335175859184329575687 \end{array} \right)$$