Prove $(1+x ) .K(x)= K\left(\frac{2 \sqrt{x}}{(1 + x) }\right)$ using Binomial Series

Question

Question

Prove $(1+x ) .K(x)= K\left(\frac{2 \sqrt{x}}{(1 + x) }\right)$ using Binomial Series.

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Notes

1. Here $K()$ is the Complete Elliptic Integral of the First Kind.

2. The argument convention here follows the wikipedia entry on Elliptic Integrals where $k$ is the elliptic modulus (rather than the paramter $m=k^2$ as used by Wolfram Alpha, Abramowitz & Stegun).

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Context

This identity can be proved simply by invoking Gauss's Transformation.

But I would like to see it proved using Binomial Series to give me some insight into solving a more complicated conjectured identity: $$(1-x ) 2E(x^2)+(x^2-1)K(x^2)=(1+x )E\left(\frac{4x}{(1 + x)^2 }\right)$$ which I am struggling with in this Stack Exchange Mathematics post

Status

Calculating a limited number of power series terms by each side of the equation and comparing them by inspection and checking their equality supports the conjecture but does not amount to a proof.

I realize that one valid proof is to show the equality of the binomial coefficients (obtained from each side of the equation) $C_r$ for the term involving $x^r$ where $r$ is any specific power of $x$ (above some small initial values of $r$ for which equality can be confirmed by calculation and inspection).

I also know that a proof can be obtained by induction - i.e. confirming equality for low value(s) of $r$ then proving equality of the differential $dC = C_{r+1}-C_r$.

However I have explored these approaches in my other SE Math question and ended up in a bit of a quagmire. So it seems that I am missing some tricks.