Answer
The proof is below.
Work Step by Step
We know from conservation of momentum that, since the masses are equal, the overall velocity in the x and y direction are equal before and after the collision. We know that initially the velocity is only in one direction (which we will define as the x direction). Since the collision isn't head on, this means that the first object has a y-component to its velocity after the collision. This means that the y-components after the collision are equal and opposite. Thus:
$vsin\theta=-vcos\phi$
$sin\theta=-cos\phi$
Where $\theta$ is the angle from the x axis of the first object and where $\phi$ is the angle from the x axis of the second object. The only time where this statement is true is if theta and phi are both 45 degrees, meaning that the two velocity vectors are perpendicular.
