Let $\displaystyle \theta$ be an angle, and let $\displaystyle m$ be the straight line through the origin with inclination $\displaystyle \theta$ to the positive $\displaystyle x$-axis. Show that the plane transformation $\displaystyle Q = f(P)$ that maps the point $\displaystyle P(x,y)$ to the point $\displaystyle Q(x',y')$ where

$\displaystyle x' = x \cos 2 \theta + y \sin 2 \theta$

$\displaystyle y' = x \sin 2 \theta - y \cos 2 \theta$

is in fact the reflection $\displaystyle \rho_m$

Note that $\displaystyle \rho_m$ is the reflection through line $\displaystyle m$

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