Mocking the J

Algebra Level 3

j=1(cosjπ2j+isinjπ2j)=?\large \prod_{j=1}^\infty \left(\cos \frac{j\pi}{2^{j}} + i \sin \frac{j\pi}{2^{j}} \right) = \, ?

Clarification: i=1 i = \sqrt{-1} .


Inspiration.

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