Problem 5.241

Demonstrate that the angle $\theta$ between the propagation direction of light and the $x$ axis transforms on transition from the reference frame $K$ to $K^{\prime}$ according to the formula $$\cos \theta^{\prime}=\frac{\cos \theta-\beta}{1-\beta \cos \theta}$$ where $\beta=V / c$ and $V$ is the velocity of the frame $K^{\prime}$ with respect to the frame $K$. The $x$ and $x^{\prime}$ axes of the reference frames coincide.