Cosine of Half Angle 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:
 * $\cos \dfrac A 2 = \sqrt{\dfrac {\sin s \sin \left({s - c}\right)} {\sin b \sin c} }$

where $s = \dfrac {a + b + c} 2$.