Napier's Analogies/Tangent of Half Difference of Sides

Napier's Analogies
Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$.

Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively.

Then:
 * $\tan \dfrac {a - b} 2 = \dfrac {\sin \frac {A - B} 2} {\sin \frac {A + B} 2} \tan \dfrac c 2$

Also see

 * Delambre's Analogies