On the Wolfram webpage, one can find the following identity for the complete elliptic integal of the first kind $K(z)$:
$$K(z) = \frac{2}{1+\sqrt{1-z}} K\Big( \Big( \frac{1-\sqrt{1-z}}{1+\sqrt{1-z}} \Big)^2 \Big)$$
See http://functions.wolfram.com/EllipticIntegrals/EllipticK/17/01/0007/.
Does anyone have a good reference for this (I checked the DLMF and Abramowitz/Stegun, but did not find anything). Or is there a simple proof that I am missing?
Thanks in advance!