Solution-ception

Algebra Level 4

If AA is the maximum value of

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

and BB is the maximum value of

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

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

Details and Assumptions

  • θ[π2,π2]\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.
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