Analogue Formula for Spherical Law of Cosines/Proof 2

Proof

 * Spherical-Cosine-Formula-Analog.png

Suppose $c$ is less than $\dfrac \pi 2$.

Let $BA$ be produced to $D$ so that $BD = \dfrac \pi 2$.

Then:
 * $AD = \dfrac \pi 2 - c$

and:
 * $\angle CAD = pi - A$

Let $C$ and $D$ be joined by an arc of a great circle, denoted $x$.

From the triangle $\sphericalangle DAC$, using the Spherical Law of Cosines:

From the triangle $\sphericalangle DBC$, using the Spherical Law of Cosines:

Hence:
 * $\sin a \cos B = \sin c \cos b - \cos c \sin b \cos A$

The case where $c > \dfrac \pi 2$ is worked similarly, but by making $D$ the point between $A$ and $B$ such that $BD$ is $\dfrac \pi 2$.