I am struggling with a problem in baby do Carmo's Differential Geometry of Curves and Surfaces (Problem 2 in Section 2.5), which says:
Let $X(\phi, \theta) = (\sin \theta \cos \phi, \sin \theta \sin \phi, \cos \theta)$ be a parametrization of the unit sphere $S^2$. Let $P$ be the plane $x = z/ \tan \alpha$, $0 < \alpha < \pi$, and $\beta$ the acute angle which the curve $P \cap S^2$ makes with the semimeridean $\phi = \phi_0$. Compute $cos \beta$.
Following do Carmo's solution to finding the curves of a coordinate neighborhood of the sphere that make a constant angle $\beta$ with the meridians $\phi = constant$ (loxodromes):
We can suppose the curve $P \cap S^2$ is the image by $X$ of a curve $\gamma(t) = (\theta(t), \phi(t))$
$$\cos \beta = \frac{\langle X_{\theta}, \gamma'(t) \rangle}{|X_{\theta}||\gamma'(t)|} = \frac{\langle (1,0), (\theta'(t),\phi'(t)) \rangle}{|X_{\theta}||\gamma'(t)|} = \frac{\theta'}{\sqrt{(\theta')^2 + (\phi')^2 \sin^2 \theta)}{}},$$
where the term $\sin^2 \theta$ (as in Baby do Carmo for the loxodromes), is to be substituted (since in our case $x = z/tan \alpha$) by some compatibility condition.
I imagine I would do this by doing, for the first and third entries of the parametrization $X$, given by $\theta$ and $\phi$, $x = z cotg \alpha$. However, I come across with $\sin \theta = \frac{\cos^2 \theta}{\cos \phi \tan \alpha}$ which, by squaring, doesn't look very nice. I am not sure this is the right approach. Any help would be appreciated.
