I'm not even sure if there is an obvious answer or not.
If an irrational expression $A$ in radicals less then $1$ is given, how to express $A$ as a finite trigonometrical expression (not a series).
For example,
$ \frac{1}{\sqrt5} = \frac{\sin\frac{\pi}{10}}{1- \sin\frac{\pi}{10}}$
Is there a general algorithm to find such expressions?
UPD: I've checked Niven's theorem on Wiki [1], but it didn't help.