So I have a work problem that I've simplified to:
\begin{cases}x = o_h - \arctan\left(\tan(a)(\cos(r) - \frac{\sin(r)}{\cos(e)})\right)\\ y = o_p + \arcsin\left(\sin(a)(\sin(r)\cos(e) - \cos(r)\right)\end{cases}
Here, $r$, $o_h$, $o_p$, $x$, $y$ are all given, I need to find $a$ and $e$. I'm capable of simplifying to get only $a$ terms or only $e$ terms, but then there'll be an unknown nested like 3 deep in trig functions and I can't get it out. I've given up on being able to solve this, but I wanted to post here, to ensure that there isn't some brilliant trig identity application that I don't know, that bails me out.
Here's my attempt to solve:
$$\begin{cases}x = o_h - \arctan\left(\tan(a)(\cos(r) - \frac{\sin(r)}{\cos(e)})\right)\\ y = o_p + \arcsin\left(\sin(a)(\sin(r)\cos(e) - \cos(r)\right)\end{cases}\\ \begin{cases}o_h - x = \arctan\left(\tan(a)(\cos(r) - \frac{\sin(r)}{\cos(e)})\right)\\ o_p + y = \arcsin\left(\sin(a)(\sin(r)\cos(e) - \cos(r)\right)\end{cases}\\ \begin{cases}\tan(o_h - x) = \tan(a)(\cos(r) - \frac{\sin(r)}{\cos(e)})\\ \sin(o_p + y) = \sin(a)(\sin(r)\cos(e) - \cos(r))\end{cases}\\ \begin{cases}\cfrac{\tan(o_h - x)}{\tan(a)} = \cos(r) - \frac{\sin(r)}{\cos(e)}\\ \cfrac{\sin(o_p + y)}{\sin(r)\cos(e) - \cos(r)} = \sin(a)\end{cases}\\ \begin{cases}\cfrac{\tan(o_h - x)}{\tan(a)} = \cos(r) - \frac{\sin(r)}{\cos(e)}\\ \arcsin\left(\cfrac{\sin(o_p + y)}{\sin(r)\cos(e) - \cos(r)}\right) = a\end{cases}\\ \begin{cases}\cfrac{\tan(a)\cos(r) - \tan(o_h - x)}{\tan(a)} = \frac{\sin(r)}{\cos(e)}\\ \arcsin\left(\cfrac{\sin(o_p + y)}{\sin(r)\cos(e) - \cos(r)}\right) = a\end{cases}\\ \begin{cases}\cfrac{\tan(a)\sin(r)}{\tan(a)\cos(r) - \tan(o_h - x)} = \cos(e)\\ \arcsin\left(\cfrac{\sin(o_p + y)}{\sin(r)\cos(e) - \cos(r)}\right) = a\end{cases}\\ \begin{cases}\arccos\left(\cfrac{\tan(a)\sin(r)}{\tan(a)\cos(r) - \tan(o_h - x)}\right) = e\\ \arcsin\left(\cfrac{\sin(o_p + y)}{\sin(r)\cos(e) - \cos(r)}\right) = a\end{cases}\\ \arcsin\left(\cfrac{\sin(o_p + y)}{\sin(r)\cos(\arccos\left(\cfrac{\tan(a)\sin(r)}{\tan(a)\cos(r) - \tan(o_h - x)}\right)) - \cos(r)}\right) = a$$
You can see from here there's just absolutely no way to extract the $a$. Am I missing anything?