Question
Prove the following statement "For Elevations which exceed or fall short of 45° by equal amounts, the ranges are equal”.

Answer

Case I: When angle of projection
$\theta=45^\circ+\alpha$
Range, $\text{R}_1=\frac{\text{u}^2\sin2(45^\circ+\alpha)}{\text{g}}$
$=\frac{\text{u}^2\cos2\alpha}{\text{g}}$
Case II: When angle of projection
$\theta=45^\circ-\alpha$
Range, $\text{R}_2=\frac{\text{u}^2\sin2(45^\circ-\alpha)}{\text{g}}$
$=\frac{\text{u}^2}{\text{g}}\sin(90^\circ-2\alpha)$
$=\frac{\text{u}^2\cos2\alpha}{\text{g}}=\text{R}_1$

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