Is it possible to manipulate $\frac{\sin(x)}{x}=\prod_{k=1}^\infty \cos(\frac{x}{2^k})$ into Euler's Product for Sinc Function? I am interested to see if the product for the sinc function that is derived through the double angle formula: $$\frac{\sin(x)}{x}=\prod_{k=1}^\infty \cos\left(\frac{x}{2^k}\right)$$
can be manipulated into Euler's Product for the sinc: $$\frac{\sin(x)}{x}=\prod_{n=1}^\infty \left(1-\frac{x^2}{\pi^2 n^2}\right)$$
I attempted to introduce the taylor series for cosine but I am left with a complicated product of an inifnite sum and I do not know where to continue further.

Hint:
$$
\begin{align}
\cos(x)
&=\frac{\frac{\sin(2x)}{2x}}{\frac{\sin(x)}x}\\
&=\prod_{n=0}^\infty\left(1-\frac{4x^2}{(2n+1)^2\pi^2}\right)
\end{align}
$$
