All angles are in radians.
(1) Without loss of generality take coordinate axes so that
$A$ is the point$(0,0,1)$, the xzplane contains the point $C$ and
the yzplane contains the point $B$.
(2) Thinking of $A$ as the North Pole then $C$ has latitude
$u$ and longitude 0 and $B$ has latitude $v$ and longitude
$\pi/2$.
(3) Find the 3D coordinates of $B$ and $C$. Where the origin O
is the centre of the sphere ${\bf OA, OB}$ and ${\bf OC}$ are
vectors of unit length.
(4) Use scalar products and vectors ${\bf OA, OB}$ and ${\bf
OC}$ to find the lengths of the arcs $AB, BC$ and $CA$ in terms of
$u$ and $v$. The required result follows.
