Suppose two successive coordinate rotations through angles $\Phi_1$ and $\Phi_2$ are carried out, equivalent to a single rotation through an angle $\Phi$. Show that $\Phi_1/2$, $\Phi_2/2$ and $\Phi/2$ can be considered as the sides of a spherical triangle with the angle opposite to $\Phi/2$ given by the angle between the two axes of rotation.
Applying the rule cosine in the spherical triangle mentioned, we have
$$\cos (\Phi/2) = \cos (\Phi_1/2) \cos (\Phi_2/2) + \sin (\Phi_1/2) \sin (\Phi_2/2) \cos(\alpha)$$
where $\alpha$ is the angle between the axes of rotation.
Now, let's think in the case that the two consecutive rotations are around the same axis, that is $\alpha = 0$. Also, let's consider an initial coordinate system where the $z$ axis is along the axis of rotation. Then, $\Phi = \Phi_1 + \Phi_2$, but the cosine rule give us
$$\cos (\Phi/2) = \cos (\Phi_1/2) \cos (\Phi_2/2) + \sin (\Phi_1/2) \sin (\Phi_2/2)$$.
$$\cos (\Phi/2) = \cos \left(\frac{\Phi_2}{2} -\frac{\Phi_1}{2} \right) = \cos \left(\frac{\Phi_2 -\Phi_1}{2} \right) $$
If $\Phi_1 = \Phi_2 = \pi / 2$, then we get $\cos(\Phi) = -1$ and $\cos(\Phi) = 1$
In view of the above, I conclude that the problem has an mistake.
