[math-fun] MathWorld EllipticK bogon?

http://mathworld.wolfram.com/CompleteEllipticIntegraloftheFirstKind.html Eqn (18) gives Out[290]= K[Sqrt[1 - k^2]] == E[k]/(k (1 - k^2)) - K[k]/k (modulus convention) which translates to Mathematica (parameter convention) Out[284]= EllipticK[1 - k^2] == EllipticE[k^2]/(k (1 - k^2)) - EllipticK[k^2]/k which doesn't work. Could there be any such relation between K, K', and only one of {E, E'}, in light of Legendre's relation? —rwg