Galileo , inhis 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.

Galileo , inhis book Two new sciences , stated that for elevations whi

1.	Galileo , inhis 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.
Answer» Solution :For a projectilelaunched with velocity `v_(@)` at an angle `theta_(@)` , therange 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 valuesof `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`.