If, for a given velocity, the maximum range is at a projection angle of 45, then there must be equal ranges for angles above and below this. Show this explicitly.

Explanation:

The range R of a projectile is given by

The maximum range occurs when . Let be the angle above or below 45°. Now let's look at the ranges brought about by these angle differences.

Case 1: Angle above 45°

We can write the range as

Case 2: Angle below 45°

We can write the range as

Note that the equations (1) and (2) are identical. Therefore, the ranges are equal if they differ from 45° by the same amount.