Prove the law of sines for the spherical triangle PQR on surface of sphere.
BACKGROUND
Suppose we have a sphere of radius 1. Let vectors $\vec{A}$, $\vec{B}$, and $\vec{C}$ be drawn from the center of the sphere, point O, to points P, Q, and R, on the surface of the sphere, respectively.
Use this already proven identity:
$|(\vec{A}\times \vec{B}) \times (\vec{A}\times\vec{C})| = |(\vec{B}\times\vec{C})\times (\vec{B}\times\vec{A})| = |(\vec{C}\times\vec{A})\times (\vec{C}\times\vec{B})|$
to prove the law of sines for the spherical triangle PQR on surface of sphere.
STEP LAST: skipping to the end of proof...
therefore:
$\Large \frac{\sin P}{\sin p} = \frac{\sin Q}{\sin q} = \frac{\sin R}{\sin r}$
which is the law of sines for spherical triangles.
(From "Schuam's outlines: Vector Analysis, second edition", Lipschutz et. al., 2009, 1959, page 37, problem 2.51)
Hints: angle is defined as $\Large \theta = \frac{s}{r}$ when the radius ($r=1$) for case of unit sphere, the angle is equal to the arc length, $\Large \theta = s$
Thus, arc length "r" is named after spherical angle "R" because its on the opposite side of spherical triangle, and is identical to "regular angle" $\angle AOB$).
Arc length "q" is named after spherical angle "Q", and is identical to "regular angle" $\angle AOC$
$\angle AOB = r$
$\angle AOC = q$
$|\vec{A}X\vec{B}| = AB\sin (\angle AOB) = AB\sin (r)$
$|\vec{A}X\vec{C}| = AC\sin (\angle AOC) = AB\sin (q)$
The length of vectors $\vec{A}$, $\vec{B}$, and $\vec{C}$ is equal to one, because the tail starts at the orgin O and the head of the vector ends on the unit sphere. Thus:
$|\vec{A}| = 1$
$|\vec{B}| = 1$
$|\vec{C}| = 1$
Thus:
$|\vec{A} \times \vec{B}| = \sin (r)$
$|\vec{A} \times \vec{C}| = \sin (q)$
Next, if you flatten the spherical triangle and take the cross product of the two arc sides with angle P in the middle you have:
$|(\vec{A} \times \vec{B}) \times (\vec{A} \times \vec{C})| = |\vec{A}\times\vec{B}|~|\vec{A}\times\vec{C}|~sin(P)$
$|(\vec{A} \times \vec{B}) \times (\vec{A} \times \vec{C})| = \sin (r) \sin (q) \sin(P)$