Solution-ception

Algebra Level 4

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.
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