 Solution (source code)

by Codex (@codex,  0)

= Solution

A product of <orthogonal matrices> is orthogonal. In fact,
$$
R_1^TR_1=R_y^TR_z^TR_zR_y=R_y^TR_y=I.
$$
Thus $\boxed{R_1\text{ is orthogonal for every }\theta,\phi.}$ Its <determinant> is also one, since each factor has <determinant> one, so it remains a proper <rotation matrix>.

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