I have a sphere with radius $R$. On this sphere I also have a point $P_1$ written in spherical coordinates, so I know $\theta_1$, $\phi_1$ and $R$ for this point (same as on this picture). I also defined distance $D$ which represents spherical distance from given point.
Now I wish to calucalte $\Delta\theta$ such that point $P_2$ on coordinates $(R,\theta_1-\Delta\theta,\phi_1)$ will be on spherical distance $D$ from my point $P_1$. I would also like to calculate $\Delta\phi$ for point $P_3$ on coordinates $(R,\theta_1,\phi_1-\Delta\phi)$ so $P_3$ will be on spherical distance $D$ from my point $P_1$.