Question

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prove that there are two angles of projection for the same horizontal range​

Answer 1

There are two angles of projection for which the horizontal range is same i.e. θ and 90o - θ.

Maximum height H1  for angle projection θ

H1 = (u sin θ)2/ 2g

Maximum height H2  for angle projection θ

H2 = (u sin90o - θ)2/ 2g = (u cos θ)2/ 2g

Now, H1+H2  = (u sin θ)2/ 2g + (u cos θ)2/ 2g = u2 / 2g.