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.