Artificial intelligent assistant

If $a$, $b$ are rotations of $S^2$ through the same angle, show that they are conjugate in $SO(3)$. Basically everything to do with the question is in the title! I'm just not really sure where to begin with this question, as I don't want to get bogged down in complicated matrix multiplication. I'm sure there must be a more geometric argument, for example because the determinant is equal to 0, $S^2$ cannot enlarge at all, but I'm not sure what other knowledge to use?

Embed $\mathbf{S}^2$ in $\mathbf{R}^3$. For each rotation has a unique vector line on which it acts as the identity, and acts as a plane rotation on the orthogonal plane to this vector line, take both such vector lines for your two rotations of same angle, and take both associated orthogonal planes. As both rotations have same angle, your two rotations have the same matrix in two (different, associated to the decompostion "fixed line$\oplus$orthogonal plane") orthonormal basis of $\mathbf{R}^3$. Then simply take the change of basis matrix for this couple of basis, and it will give you the matrix realizing conjugation between you two rotations, and ensure that this matrix is in $SO(3,\mathbf{R})$, which is always possible.

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