Constant latitude path vs great circle path (geodesic) on a sphere

Question

How is it that on sphere S with spherical coordinates the path of constant latitude $\gamma(t)=(\phi(t),\theta(t)=(t,\theta_0))$, $t$ is from the interval $[\phi_a,\phi_b]$ is not a geodesic. The length of this path is $Rsin(\theta_0)\left|\phi_b-\phi_a\right|$, whereas the length of the great circle path (geodesic) connecting these two points is $R\left|\phi_b-\phi_a\right|$ that is obviously longer.

Answer 1

Your claim on the length of the great circle path is quite wrong. You need to find the angle the two points subtend in the great circle passing through those points. HINT: Dot the two vectors from the center of the sphere to the respective points.