Solution
= Solution
Let
$$
J_-=\begin{pmatrix}0&-1\\1&0\end{pmatrix},
\qquad
J_+=\begin{pmatrix}0&1\\1&0\end{pmatrix}.
$$
Since $J_-^2=-I$ and $J_+^2=I$, separating the even and odd terms of the <matrix exponential> gives
$$
\boxed{R=e^{\theta J_-}
=\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}},
$$
and
$$
\boxed{S=e^{\theta J_+}
=\begin{pmatrix}\cosh\theta&\sinh\theta\\\sinh\theta&\cosh\theta\end{pmatrix}}.
$$