Distance (r) as the angle between two stars are calculated by following formula: $$ \cos(r) = \sin(\delta_1) \sin(\delta_2) - \cos(\delta_1)\cos(\delta_2)\cos(\alpha_1-\alpha_2) $$ $\delta$ and $\alpha$ are the sky coordinates of the stars and they have errors as $\Delta \delta_{1,2}$ and $\Delta \alpha_{1,2}$. I want to calculate error in distance ($\Delta r$) by using the errors in coordinates. But I could not find appropriate solution. First I calculated the derivative of this formula; $$ \begin{align} -\Delta r \sin(r) =& \Delta \delta_1[\cos(\delta_1)\sin(\delta_2) - \sin(\delta_1)\cos(\delta_2)\cos(\alpha_1-\alpha_2)] \\ +& \Delta \delta_2[\cos(\delta_2)\sin(\delta_1) - \sin(\delta_2)\cos(\delta_1)\cos(\alpha_1-\alpha_2)] \\ -& \Delta(\alpha_1-\alpha_2)\cos(\delta_1)\cos(\delta_2)\sin(\alpha_1-\alpha_2) \end{align} $$ but couldn't see the next step. I took $\Delta(\alpha_1-\alpha_2)= \sqrt{\left(\Delta\alpha_1\right)^2 + \left(\Delta\alpha_2\right)^2}$ in here.
Does anyone have knowledge about this?