Formula for the fourth side of a spherical quadrilateral

After some manipulations, the nicest formula I've found so far is

$$\begin{align*} \cos d = \qquad &\cos a \cos b \cos c\\ +\ &\sin b\,\left(\sin a \cos c \cos \theta_{ab} + \cos a \sin c \cos \theta_{bc}\right)\\ +\ &\sin a \sin c\, \left(\sin \theta_{ab}\sin \theta_{bc} - \cos b \cos \theta_{ab} \cos \theta_{bc} \right).\end{align*}$$

I haven't checked my work too closesly, but the above does have the right symmetries, and reduces to the right formulas as any side length goes to zero.

I'm still hoping there are simplifications of the above, or a way of removing the sines of $\theta_{ab}$ and $\theta_{bc}$.