For elevations which exceed or fall short of 45° by equal amounts . . .

Question : For elevations which exceed or fall short of 45° by equal amounts, the ranges are equal”. Prove this statement.

Doubt by Yana and Shruti 

Solution : 

Case I : 
θ₁=45°+α
We know, 
Range (R) = u²sin2θ/g

R₁=u²sin2θ₁/g
R₁=u²sin2[45°+α]/g

R₁=u²sin[90°+2α]/g

R₁=u²(cos2α)/g [∵sin(90°+θ)=cosθ]

R₁=u²(cos2α)/g — (1) 

Case II : 
θ₁=45°-α

R₂=u²sin2θ₂/g
R₂=u²sin2[45°-α]/g

R₂=u²sin[90°-2α]/g

R₂=u²(cos2α)/g [∵sin(90°-θ)=cosθ]

R₂=u²(cos2α)/g — (2) 

From (1) and (2), it is clear that when angle of elevations exceed or fall short of 45° by equal amounts, the ranges are equal.

Hence statment is proved successfully.