Forgive me if this question goes beyond the realms of trig, I have only placed it here for lack of a better category.
My difficulty is with the following problem: cot(x) = x. Solve for x.
This is a little problem that I have been contemplating for quite a while. After spending a fairly ridiculous amount of time trying to solve it (by algebra, trig identities and the like), and showing it to many other people who know much more math than I, I have come to the determination that it is simply not solvable by algebra. Now of course, a very good approximation could be obtained using Newton's method or perhaps even the Taylor series. But unfortunately these are only approximations of the exact answer. Is there any way to obtain an exact answer for this equation? Do we even have a way of notating the exact answer? And if not can it be proved that an exact answer cannot be determined? Thank you so much for any help that any of you have to offer!