Complex Product Into Complex Sum

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

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