<s>[INST] Can $\prod\limits_{k=0}^n \left( 2 \cosh(2^kx)-1 \right)$ be simplified? Do you know if the product $\prod\limits_{k=0}^n \left( 2 \cosh(2^kx)-1 \right)$ can be simplified?
 [/INST] Note that
$$\begin{align*}
2\cosh(2^k x)-1 &= \exp(2^k x) - 1 + \exp(-2^k x) \\
&= \exp(-2^k x) \frac{\exp(3\cdot2^k x) + 1}{\exp(2^k x) + 1}\\
&= \frac{\cosh(\frac{3}{2}2^k x)}{\cosh(\frac{1}{2}2^k x)}.
\end{align*}$$
Then we have
$$\begin{align*}
\prod_{k=0}^{n} \left(2\cosh(2^k x)-1 \right)
&= \prod_{k=0}^{n}\frac{\cosh(\frac{3}{2}2^k x)}{\cosh(\frac{1}{2}2^k x)} \\
&= \frac{\sinh(\frac{3}{2}2^{n+1} x)/\sinh(\frac{3}{2} x)}{\sinh(\frac{1}{2}2^{n+1} x)/\sinh(\frac{1}{2} x)} \\
&= \frac{2\cosh(2^{n+1}x)+1}{2\cosh x+1}.
 \end{align*}$$
Sorry for this unfriendly solution. I typed these formulas with my iPad3 and this drove me nearly mad.
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