For $x>1$, evaluate $\lim_{n\to \infty} \prod _{k=0} ^n \left(1+\frac{2}{x^{2^k} + x^{-2^k}}\right)$ Simplifying the part after the product function
For $k=0$
$$\frac{x+x^{-1} +2}{x+x^{-1}}$$
$$=\frac{(x^{\frac 12}+x^{-\frac 12})^2}{x+x^{-1}}$$
Performing a similar simplification for all values of $k$, then end result is
$$\frac{1}{x^{2^k}+x^{-2^k}} (x^{\frac 12}+x^{-\frac 12})^2 (x+x^{-1})(x^2+x^{-2})...(x^{2^{k-1}}+x^{-2^{k-1}})$$
I am not able to simplify it further. How should I proceed?

Hint :
Multiply and divide by $(x^{\frac{1}{2}}-x^{−\frac{1}{2}})$
and use $(a+b)(a-b)=a^2-b^2$.
