= Solution
Choose orthonormal coordinates with origin $P$ and horizontal axis $l$. The <Euclidean reflection> and <planar rotation> have <matrices>
$$
S=\begin{pmatrix}1&0\\0&-1\end{pmatrix},\qquad R_\alpha=\begin{pmatrix}\cos\alpha&-\sin\alpha\\\sin\alpha&\cos\alpha\end{pmatrix}.
$$
Their <function composition> therefore satisfies the <composition of a plane rotation and a reflection> identity
$$
R_\alpha S=\begin{pmatrix}\cos\alpha&\sin\alpha\\\sin\alpha&-\cos\alpha\end{pmatrix}=R_{\alpha/2}S R_{-\alpha/2}.
$$
Conjugating by a <planar rotation> transports the horizontal reflection axis to the line at angle $\alpha/2$. Equivalently, the vector $e=(\cos(\alpha/2),\sin(\alpha/2))$ is fixed, whereas its perpendicular $e_\perp=(-\sin(\alpha/2),\cos(\alpha/2))$ is sent to $-e_\perp$. Consequently \b[the fixed line is the line through $P$ obtained by rotating $l$ through $\alpha/2$, and $\tau\rho$ is reflection in that line]. Angles of an unoriented line are understood modulo $\pi$, so this description is independent of the representative chosen for $\alpha$ modulo $2\pi$.
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