I am trying to find a spherical parameterization of the great circle path on a sphere of radius R. I parameterize my sphere as follows:
$(R\sin\theta\cos\phi, R\sin\theta\sin\phi, R\cos\theta)$.
I know that the great circle is the intersection of a plane passing through the origin. The equation of such a plane is $ax+by+cz = 0$.
According to this article http://sgovindarajan.wikidot.com/twosphere, by substituting $x,y,z$ into the equation of my plane I should be able to obtain:
$\cot\theta = c_1(\cos(\phi + c_2))$
where $c_1$ and $c_2$ are simply constants. However, I cannot seem to see how the $\sin \phi$ disappears. I began by isolating the $\theta$'s and $\phi$'s. But I cannot seem to arrive at the above result. Is there a trigonometric property I should be using?