Delambre's Analogies/Cosine by Cosine

Delambre'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:
 * $\cos \dfrac c 2 \cos \dfrac {A + B} 2 = \sin \dfrac C 2 \cos \dfrac {a + b} 2$

Proof
In the below, we have:
 * $s = \dfrac {a + b + c} 2$

Thus:

Also see

 * Napier's Analogies