Is there any reference providing a derivation or a statement of an identity $$ \text{Re}\ \text{K}(x) = \frac{1}{1+x}\text{K}\left(\frac{2\sqrt{x}}{x+1}\right), \quad 0 < x < \infty , $$ where $$ \text{K}(x) =\int \limits_{0}^{\frac{\pi}{2}}\frac{dt}{\sqrt{1-x^2\sin^2(t)}} $$ is the elliptic integral of the first kind?
Although this identity is simply derived (look here) and can be verified graphically (in Wolfram Mathematica), I can't find any reference in which it is stated. This is very strange, leading to worries about its correctness.
See Erdelyi et al., Higher Transcendental Functions, Vol II, p.319, Table 4, last entry. The book is available via http://en.wikipedia.org/wiki/Bateman_Manuscript_Project