I am trying to algebraically prove that multiplying 2 quaternion rotations produces a new quaterion rotation. This is what I have so far:
After lots of simplifying, I get here:
I see the first term has the cosine part of the combined quaternion. I almost have the second term of rodrigues rotation formula in the second term. I am having a hard time figuring out how I can simplify down to the the rest of the rodrigues rotation formula to show that applying 2 quaternions is equivalent to applying the composed quaternion.