I want to find the necessary conditions for positive sequence of real numbers, $\{a_n\}$ for which the following difference equation has a solution where $0 < \theta_n < \frac{\pi}{4}, \forall n$.
$$\theta _{n+1} = \arctan\left(\frac{a_{n+1}}{a_n}\tan(2\theta_n)\right)$$
All I know about how to deal with such problems is "Oscillation Theory" (e.g. Saber Elyadi's Introduction to Difference Equations) where we have a set of conditions for nonoscillatory solutions of difference equations of the form $x_{n+1} - x_n + p(n)x_{n-k} = 0$ but it is not obvious how to extend those to trigonometric functions and even change of variables (e.g. using $x_{n+1} = \frac{2a_{n+1}}{a_n} \times \frac{x_n}{1-x_n^2}$) only yield nonlinearities that I cannot deal with. Any reference to the literature is also helpful.