7.53. A light ray falls on a flat mirror at an angle $\alpha$. The mirror is rotated by an angle $\beta$ around the projection of the ray on the mirror. By what angle will the reflected ray deviate in this case?
7.53. Let $B$ be the point of incidence of the ray on the mirror; $A$ - a point on the ray, different from $B$; $K$ and $L$ - the projections of $A$ on the mirror in its initial and new positions; $A_{1}$ and $A_{2}$ - points symmetric to $A$ relative to the initial positions of the mirror. The required angle is equal to the angle $A_{1} B A_{2}$. If $A B=a$, then $A_{1} B=A_{3} B=a$ and $A K=$ $=a \sin \alpha$. Since $\angle K A L=\beta$, then $A_{1} A_{2}=2 K L=2 A K \sin \beta=$
$=2 a \sin \alpha \sin \beta$. Therefore, if $\varphi$ is the required angle, then $\sin (\varphi / 2)=\sin \alpha \sin \beta$.