Some R2 complex exponential calculations using CliffordBasic.

<< CliffordBasic`;
$SetSignature={2,0};

ClearAll[ a, b,c, angle, exp, r ]
exp[ t_ ] := Module[ {abs},
abs = Sqrt[-GeometricProduct[t,t]];
Cos[ abs ] + (t/abs) Sin[ abs ]];
r[t_] := GeometricProduct[ e[1], exp[t ] ];
a =  r[e[1,2] Pi/3]
b  =  r[e[1,2] Pi/6]
c = GeometricProduct[ a,b] // Simplify
angle = exp[ e[1,2] Pi(-1/3 + 1/6)]
iangle = MultivectorInverse[angle]// Simplify
GeometricProduct[ c,iangle ]

{Cos[Pi/3],Sin[Pi/3]}
{Cos[Pi/6],Sin[Pi/6]}