Cosine of Half Side for Spherical Triangles

Theorem
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:
 * $\cos \dfrac a 2 = \sqrt{\dfrac {\cos \left({S - B}\right)S \cos \left({S - C}\right)} {\sin B \sin C} }$

where $S = \dfrac {A + B + C} 2$.