Tangent of Half Side 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:
 * $\tan \dfrac a 2 = \sqrt {\dfrac {-\cos S \, \map \cos {S - A} } {\map \cos {S - B} \, \map \cos {S - C} } }$

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

Proof
Hence the result.

Also see

 * The other Half Side Formulas for Spherical Triangles:
 * Sine of Half Side for Spherical Triangles
 * Cosine of Half Side for Spherical Triangles


 * Half Angle Formulas for Spherical Triangles:
 * Sine of Half Angle for Spherical Triangles
 * Cosine of Half Angle for Spherical Triangles
 * Tangent of Half Angle for Spherical Triangles