Given any right triangle $FGH$ with right angle at vertex $G$. A line $L$ intersects $G$ at some angle $\theta$ relative to short side GF (assume acute angle). Extend $HF$ so it intersects line $L$ at $K$. What is the length of $GK$ in terms of length $GF$ (that is, $GK$ projected onto line $L$ as viewed from vertex $H$)?
At first I thought dividing $GF$ by $\cos(\theta)$ or $\tan(\theta)$ would work, but no. This is not the usual projection, that is, using a perpendicular to $L$ from $F$. After much thought but no idea what to do, I'm stumped.
I'm trying to derive the sum of $\arctan(x) + \arctan(y)$ identity (I have not included the drawing that would pertain directly to that), but I need to solve this part of the problem first.