The equality $$K(i)E(i)-K(i)^2=\pi/4\tag 1$$ follows from $$\int_0^{\pi/2}\frac{\sin^2\theta}{\sqrt{1-k^2\sin^2\theta}}d\theta=\frac{K(k)-E(k)}{k^2}$$ and evaluation of the integral when $k^2=-1$ by converting the integral into a beta function integral.
Can we show the equality $(1)$ in some other ways? Is there a geometric meaning of this identity?
Thanks in advance.
Note: WA is using $m=k^2$ convention.