Answer
$V_{T} \times cot\theta$
Work Step by Step
In the reference frame of the train, the rain is moving left and downwards. The speed of the the rain in the reference frame of the Earth is the speed that the rain is moving downwards in the reference frame of the train. Therefore we need to calculate the speed of the rain moving downwards. Let the vector $V_{D}$ be the vertical velocity, $V_{L}$ be the horizontal velocity, then $\theta$ is the angle made by $V_{L}$ and the result velocity as shown in the picture. Therefore $V_{D}$ = $V_{T} \times cot\theta$
