Galileo, in his book Two new sciences, stated that "for elevations which exceed or fall short of `45^(@)` by equal amounts, the ranges are equal. Prove this statement.

Explanation

For a projectile launched with velocity `v_(0)` at an angle `theta_(0)`, the range is given by
`R=(V_(0)^(2)sin2theta_(0))/g`
Now, for angles, `(45^(@)-alpha)`, `2theta_(0)` is `(90^(@) + 2alpha)` and `(90^(@)-2alpha)`, respectively. The value of `sin(90^(@)+2alpha)` and `sin(90^(@)-2alpha)` are the same, equal to that of `cos 2alpha`. Therefore, ranges are equal for elavations which exceed or fall short of `45^(@)` by equal amounts `alpha`.