= Solution
The standard isomorphisms are
$$
SO(2)\longrightarrow B,
\qquad
\begin{pmatrix}\cos\theta&-\sin\theta\\
\sin\theta&\cos\theta\end{pmatrix}
\longmapsto e^{i\theta},
$$
and
$$
\mathbb R/\mathbb Z\longrightarrow B,
\qquad
t+\mathbb Z\longmapsto e^{2\pi it}.
$$
For $z\in B$, let $R_z$ be the corresponding rotation. The identity
$$
JR_zJ=R_{z^{-1}}
$$
shows that
$$
(z,\epsilon)\longmapsto
\begin{cases}
R_z,&\epsilon=1,\\
R_zJ,&\epsilon=-1
\end{cases}
$$
respects the multiplication of the <generalized dihedral group>. The decomposition in part (b) makes it bijective. Hence
$$
\boxed{O(2)\cong D(B)}.
$$
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