In the picture bellow; How to prove that the result of rotation about $P$ through angle $\theta$, followed by rotation about $Q$ through angle $\varphi$ is rotation about $R$ through some angle? ــ Can it be proven by the reflections of the great circles $\mathscr{L,M,N}$? How?
How to find the axis of the product rotation?
The first rotation is the reflection through $L$ followed by reflection through $M$. For the second, it's $M$ followed by $N$. The $M$'s cancel, and you're left with a rotation with axis $OR$.