Are we in sinc?

Calculus Level 4

$\large \cos \left(\frac{\pi}{12}\right) \cos \left(\frac{\pi}{24} \right)\cos \left(\frac{\pi}{48} \right) \cos \left(\frac{\pi}{96}\right) \cdots = \ ?$

Give your answer to 3 decimal places.

Bonus: Can you find the exact value of the expression?