We begin by stating, without proof, that the equation for induced drag CDi over a wing with circulation Γ(y) in a flow with density ρ∞ and a downwash velocity of w(y). y represents position on the wing measured from the root, varying from -b/2 to +b/2, where b is wingspan.
Di = -∫-b/2+b/2ρ∞w(y)Γ(y)dy
Next, we take the elliptical distribution of circulation around a wing, which depends on y:
Γ(y) = Γ0√(1 - (y/(b/2))2)
Finally, the downwash velocity for this planform simplifies to:
w(y) = -Γ0/(2b)
We may now substitute into the integral expression:
Di = -∫-b/2+b/2ρ∞(-Γ0/(2b))(Γ0√(1 - (y/(b/2))2))
We introduce the coordinate transfomation y = -s cos φ, and simplify the expression to:
Di = (ρ∞Γ02/(2b))∫0π(b/2) sin φ√(1 - cos2 φ) dφ
Di = (π/8)ρ∞Γ02
We now apply the formula for the drag coefficient:
CDi = Di/(0.5*ρ∞V∞2S)
CDi = πΓ02/(4V∞2S)
This is all well and good, but the circulation at the wing root is not a practical design variable. We then state, without proof, that based on the expression for the lift coefficient of the same wing:
Γ0 = 2CLV∞S/(πb)
Thus, we can substitute into our expression for the drag coefficient:
CDi = {π/(4V∞2S)}{2CLV∞S/(πb)}2
CDi = CL2 / (π*AR)
We have now derived the coefficient of induced drag for a wing with an elliptical circulation, and thus, elliptical lift distribution and planform (these follow directly from the Kutta-Joukowski lift theorem and the proportionality of chord to circulation).
Now, *poof* I present the formula for the coefficient of induced drag in general:
CDi = CL2 / (π*AR*e)
where the airplane efficiency factor e ≤ 1. Note then that for the elliptical planform, e = 1. Therefore, a wing with an elliptical planform is the most efficient (provided aspect ratio and lift coefficient are fixed), because it has the least possible value for its coefficient of induced drag.