Finding angle of intersection between two curves

I am trying to solve a problem in Do Carmo, a book which I often find incomprehensible.

Let $$X(\varphi,\theta)=(\sin(\theta)\cos(\varphi),\,\sin(\theta)\sin(\varphi),\,\cos(\theta))$$ be a parametrization of the unit sphere $$S^2$$. Let $$P$$ be the plane $$x=z\cot(\alpha)$$, $$0<\alpha<\pi$$, and $$\beta$$ be the acute angle which the curve $$P\cap S^2$$ makes with the semimeridian $$\varphi=\varphi_0$$. I am trying to compute $$\cos(\beta)$$.

I know I have to find a parametrization for both curves. First, how would I go about finding a parametrization $$\gamma$$ for $$P\cap S^2$$? The parametrization for the semimeridian I believe would be $$\alpha(\varphi):=X(\varphi,\theta_0)$$. Then once I have a parametrization for $$P\cap S^2$$ I would compute $$\cos(\beta)=\frac{\langle\alpha'(0),\,\gamma'(0)\rangle}{|\alpha'(0)|\,|\gamma'(0)|}.$$ Is this correct as well?

You should follow the calculation on p. 96 (of my edition, anyway), the loxodrome calculation, with one change. Instead of $$\theta'$$ and $$\phi'$$ being unknowns to be solved for, they are related by $$\tan\theta\cos\phi = \cot \alpha$$. Differentiate that to get a relationship between $$\theta'$$ and $$\phi'$$. Then you don't know what $$\theta'$$ and $$\phi'$$ are (because you don't have a parametrization of the curve) but you can determine the direction of the vector (i.e., up to a constant). That is good enough because we don't need unit vectors in the formula for the cosine of the angle between two vectors.
After a bit of work I get the formula $$\cos\beta = \sin\phi\sin\alpha$$. I'm not absolutely sure it's correct but it passes a few basic tests. E.g., if $$\phi=0$$ or $$\pi$$, $$\beta=\pi/2$$ (the prime meridian cuts all these circles at right angles), and if $$\phi=\pi/2$$ then we have $$\cos\beta=\sin\alpha$$ which makes sense geometrically. ($$\alpha$$ and $$\beta$$ should be complementary angles in this case. I wish doCarmo would use colatitude instead of latitude in $$x=z\cot\alpha$$.)
This seems like an obvious exercise in spherical geometry, but I can't find verification of the result anywhere (which is what led me here). I'm going to use more general spherical trigonometry with the three great circles $$x=z\cot\alpha$$, $$\phi=\phi_0$$, and the prime meridian to verify the result. I think it will work.