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_(@)` at an angle `theta_(@)` , the range is given by
`R=(v_(0)^(2)sin2theta_(0))/(g)`
Now , for angles , `(45^(@)+alpha)and(45^(@)-alpha),2theta_(0)` is `(90^(@)+2alpha)and(90^(@)-2alpha)`, respectively . The values of `sin(90^(@)+2alpha)andsin(90^(@)-2alpha)` are the same , equal to that of `cos2alpha`. Therefore , ranges are equal for elevations which exceed of fall short of `45^(@)` by equal amounts `alpha`.