Mocking the J

$\large \prod_{j=1}^\infty (\cos \frac{j\pi}{2^{j}} + i \sin \frac{j\pi}{2^{j}} )\\ \large =\prod_{j=1}^\infty e^{i\frac{j\pi}{2^{j}}}~~~~(\color{#302B94}{* })$ $\Large =e^{i \pi \large{(\displaystyle \sum_{j=1}^{\infty} \frac{j}{2^j})}}$ $\large{ =e^{2i\pi}=\huge\boxed {1}}~~~~( \color{#302B94}{** })$

$\color{#3D99F6}{\boxed{\color{forestgreen}{\boxed{\small {\color{#302B94}{* \text{Use }\cos x+ i\sin x=e^{ix}}} \\ \small {\color{#302B94}{** \text{Note: }\displaystyle \sum_{j=1}^{\infty} \frac{j}{2^j} \text{ denotes the sum of an} \\ \text{ infinite Arithmetic-Geometric Progression}\\ \text{Geometric Progression: first term as well as common ratio is }\frac{1}{2} \\ \text{Arithmetic Progression: first term as well as common difference is } 1 }}}}}}$

Partial Credits: To Rishabh Cool, for the LaTeX.