I am trying to calculate the optimal flight climb angle based off of some basic power calculations for an aircraft on steady climb with an angle of theta. Doing a balance of forces I obtained the thrust force, then multiplied by velocity to obtain power. The power was time independent, thus multiplying the term for power by time I obtained the energy, and then I try to optimize this by trying to find an angle for which energy is minimized, however it seems only pi/2 radians (90 degrees) gives a minimum on the graph of energy consumption which I obtained. Did I do something wrong? Is my approach wrong or is the question in itself flawed? I attach my calculations below as images, thank you!
PS: This is not for homework, this is an interest of mine, I am on my second year studying aerospace engineering and am doing this (studying the efficiency/power equations of a UAV) as a way of getting 'ahead of the curve'. Any feedback on my method is appreciated :)