Four-Parts Formula/Corollary

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

Proof
Let $\triangle A'B'C'$ be the polar triangle of $\triangle ABC$.

Let the sides $a', b', c'$ of $\triangle A'B'C'$ be opposite $A', B', C'$ respectively.

From Spherical Triangle is Polar Triangle of its Polar Triangle we have that:
 * not only is $\triangle A'B'C'$ be the polar triangle of $\triangle ABC$
 * but also $\triangle ABC$ is the polar triangle of $\triangle A'B'C'$.

We have: