Evaluating $\prod\limits_{n=1}^{\infty}(1+e^{-x2^n})$

Question

In this page, it is stated that $$ \prod\limits_{n=1}^{\infty}\left(1+e^{-x2^n}\right) = \frac{1}{2}(1+\coth(x)) $$ How can one show this? In the webpage it is under the title "Euler's product", which I understand has something to do with representing the Riemann zeta function as an infinite product.

Answer 1

One may recall that $$ \prod\limits_{n=1}^\infty\left(1+z^{2^n}\right)=\frac1{1-z^2},\quad |z|<1, \tag1 $$ which may be proved by observing that $$ (1-z^2)\prod\limits_{n=1}^N\left(1+z^{2^n}\right)=(1-z^{2^{N+1}}) \tag2 $$ then by putting $z=e^{-x}$ in $(1)$ ($x>0$) one may express $\dfrac1{1-e^{-2x}}$ in terms of $\coth x$.

Can you take it from here?