So I bought this book in the library and physics fascinates me and I found this quote in the book " Galileo has proved that when any effects due to air resistance are ignored, the ranges for projectiles (on a level field) whose angles of projection exceed or fall short of 45° by the same amount are equal.This can be easily proved" I don't understand it at all? And I don't think it is that easy to prove, but I want to know it so much :/ Can you explain this to me?
If the projectile is launched at an angle $\theta$ the range is proportional to $\sin(2\theta)$, and the function $y = \sin(2\theta)$ is symmetric about $\theta = 45^\circ$ i.e. $\sin(2(45 - \delta)) = \sin(2(45 + \delta))$ for any angle $\delta$. That's why the range is the same whether you add or subtract the angle $\delta$ to your 45$^\circ$ launch angle.