I've shown using Euler-Lagrange equation that the shortest path between two points on a unit sphere is $$\phi = -\arcsin\left(\frac{\cot \theta}{\sqrt{\frac{1}{c_1^2}-1}}\right) + c_2$$
Now, I need to show that $\phi$ satisfies $$x\sin(c_2) + y\cos(c_2) - \frac{z}{\sqrt{\frac{1}{c_1^2}-1}} = 0$$ where $x=\sin(\theta)\cos(\phi)$, $y=\sin(\theta)\sin(\phi)$ and $z=\cos(\theta)$ as usual on the unit sphere with $R=1$.
I did the following: $$\sin(\phi - c_2) = \frac{-\cot(\theta)}{\sqrt{\frac{1}{c_1^2}-1}}$$ $$\sin(\phi)\cos(c_2) - \cos(\phi)\sin(c_2) = \frac{-\cos(\theta)}{\sin(\theta)\sqrt{\frac{1}{c_1^2}-1}}$$ $$\sin(\theta)\sin(\phi)\cos(c_2) - \sin(\theta)\cos(\phi)\sin(c_2) = \frac{-\cos(\theta)}{\sqrt{\frac{1}{c_1^2}-1}}$$
$$y\cos(c_2) -x\sin(c_2) +\frac{\cos(\theta)}{\sqrt{\frac{1}{c_1^2}-1}} = 0$$ Problem is that I'm missing a minus sign before $y\cos(c_2)$ and I can't find where I went wrong.
Thanks