Proof of angular velocity identity

I've been going thru the book by Hodges and Roithmayr regarding dynamics theory and I can't seem to prove that:

I have tried using multiple substitutions, including the fact that every component of the direction cosine matrix is equal to its cofactor, in multiple, different ways. But I feel like I've been walking in circles. The last attempt I made was, when using the definition of angular velocity:

And then writing every derivative of the coordinate system by substituting its direction cosines and substituting the cofactors inside. This led to another interesting thing:

Which may or may not be related to the proof I'm looking for, but I cannot work out how. My guess is that this would be a pretty standard equation to use, since angular velocity is such a big thing in dynamics, but I could not find it anywhere.

• I think the "Maxwell–Betti reciprocal work theorem" may help. en.wikipedia.org/wiki/Betti%27s_theorem
– r13
Apr 9, 2021 at 23:19
• Could you please explain the $^A\vec\omega^B$ notation? I've never came across it. Apr 10, 2021 at 5:39
• @NMech is the angular velocity of a body B in a reference frame A. Apr 10, 2021 at 13:42
• @r13 I have found the answer, should I post it myself? Apr 10, 2021 at 13:43
• I don't know if the rule permits that (answer the question of yourself) or not. But since nobody has provided a response with a pointed answer, I think it is considered a good service to the readers who have the same question. Also, it might draw further responses with great insight. So why not, I support posting your findings.
– r13
Apr 10, 2021 at 13:52