The angle of projection for which the maximum height of the projectile is equal to the horizontal range has to be determined.
Let the required angle of projection be A and object is projected at V m/s.
The horizontal component of the velocity is V*cos A and the vertical component is V*sin A. There is only an acceleration equal to g acting vertically downwards on the object. The horizontal component of the velocity does not change.
At the maximum height the vertical velocity is equal to 0. This happens at time t where V*sin A - g*t = 0
=> t = V*sin A/g
The particle travels for a time t = 2*V*sin A/g and the horizontal range is equal to (2*V*sin A/g)*V*cos A
The maximum height is V*sin A*V*sin A/g - (1/2)*g*(V*sin A/g)^2
=> (V*sin A)^2/g - (1/2)*(V*sin A)^2/g = (1/2)*(V*sin A)^2/g
As this is equal to the range:
(1/2)*(V*sin A)^2/g = (2*V*sin A/g)*V*cos A
=> (1/2)*sin A = 2*cos A
=> sin A/cos A = 4
=> tan A = 4
=> A = 75.96 degrees
If the angle of projection is 75.96 degrees the maximum height is equal to the horizontal range.