Solution-ception

If $A$ is the maximum value of

$\displaystyle \sqrt { \cos { \theta } +\sqrt { \cos { \theta } +\sqrt { \cos { \theta } +\sqrt { \cos { \theta } +\dots } } } }$

and $B$ is the maximum value of

$\displaystyle \sqrt { \sin { \theta } +\sqrt { \sin { \theta } +\sqrt { \sin { \theta } +\sqrt { \sin { \theta } +\dots } } } },$

what is the value of $2A-B?$

Details and Assumptions

• $\theta \in \left[-\frac{ \pi}{2},\frac{\pi}{2}\right]$

• Round your answer to four significant figures.

You might also want to try Generalized infinite square roots , as it was the inspiration for this problem.