Spherical Law of Tangents

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:
 * $\dfrac {\tan \frac 1 2 \left({A + B}\right)} {\tan \frac 1 2 \left({A - B}\right)} = \dfrac {\tan \frac 1 2 \left({a + b}\right)} {\tan \frac 1 2 \left({a - b}\right)}$

Also known as
This result is also known as the Spherical Law of Tangents.