The above expression can be simplified to:
\(\sqrt{\cfrac{1+\sqrt{\cfrac{1-\sqrt{\cfrac{1-\sqrt{\cfrac{1+\cfrac{\sqrt{3}}{2}}{2}}}{2}}}{2}}}{2}} = \sqrt{ \frac{1 + \cos\left(\frac{13\pi}{48} \right)}{2}} \)
The above value is the cosine of \(13\pi/96\)
So the answer to the question is: \(13\pi/96\)
-Vinculum