For a more intrinsic perspective, parametrize the sphere as $$\varphi(u,v)=(\sin(u)\cos(v),\sin(u)\sin(v),\cos(u)),$$
so that the coefficients of the first fundamental for are $E=1$, $F=0$ and $G=\sin^2(u)$. Then a lattitude circle on the sphere is a $v$-curve associated with this parametrization, and thus may be parametrized as $$\alpha_v(t)=\varphi(u_0,t)=(\sin(u_0)\cos(t),\sin(u_0)\sin(t),\cos(u_0)),$$
with $0<u_0<\pi$. Its arclength parametrization is then given by $$\beta(s)=\varphi\left(u_0,\frac{s}{\sin(u_0)}\right).$$
Now when $F=0$ there is a nice formula for the geodesic curvature $\kappa_g$ of an arclength parametrized curve of the form $\varphi(u(s),v(s))$ (for a general parametrization $\varphi$ of a regular surface), namely, $$\kappa_g=\frac{1}{2\sqrt{EG}}(G_uv'-E_vu')+\theta',$$ where $\theta$ is the angle from $\varphi_u$ to the velocity vector of the curve, where $\varphi_u=\frac{\partial}{\partial u}\varphi$. In our case, the velocity vector $\beta'(s)$ is precisely $\varphi_v=\frac{\partial}{\partial v}\varphi$, which is always orthogonal to $\varphi_u$ since $F=0$, thus $\theta'=0$. Moreover, we have $u'=0$, $v'=1/\sin{u_0}$ and $G_u=2\sin(u_0)\cos(u_0)$, thus $$\kappa_g=\cot(u_0).$$