Since the idea is quite different, this is not tacked on to my previous answer.
There may be a connection to the idea of mixedmath. The motivation was actually Ross Millikan's sketch of a calculation of the fourth root of a rotation matrix.
Suppose that we are explicitly given a rotation matrix, where for simplicity the rotation angle $\theta$ is between $0$ and $\pi$. We want to find the matrix for a rotation by $\theta/2$, $\theta/4$, or more generally $\theta/2^k$.
Compute what the rotation does to $(1,0)$. Say the result is $(a,b)$.
Find the sum $(1,0)+(a,b)=(1+a,b)$. Divide by the norm. This tells us where the point (1,0) is taken by rotation through $\theta/2$. Now we can immediately write down the appropriate rotation matrix.
For the rotation through $\theta/4$, repeat. For rotation through $\theta/2^n$, repeat more often.
We could think of this process as using the half-angle formulas for $\sin$ and $\cos$. That's certainly not the way I got to it, the idea was entirely primitive geometric. I prefer to think of the half-angle formulas as consequences of the above-described process.