Please see attached image.

1) For what value(s) of b does

$\displaystyle \sqrt{\frac{A_1}{A_2}}=2b-1$ for all $\displaystyle \theta$ ?

(A1=area of the small triangle on top, A2=area of the bigger triangle on the bottom with base b)

2) bonus problem:

Given the foregoing, express $\displaystyle \frac{A_1}{A_2}$ as a hyperbolic function with b in the argument. Express b as trig function with theta in the argument.

I would never be able to solve this if I hadn't been thinking about it for the past few months.