It is known that hypergeometric functions are closely related to the formula of $\pi$ given by Ramanujan. Trying to master the proof given by the Borwein brothers, I got two identities:
$$\mathrm{A}:(1+k^2)\left(\frac{2K(k)}{\pi}\right)^2={}_3 F_2\left(\begin{matrix}1/4& 3/4& 1/2\\1& 1\end{matrix};\frac{16k^2(1-k^2)^2}{(1+k^2)^4}\right)$$
with $0\leq k\leq\sqrt{2}-1$ and
$$\mathrm{B}:(\frac{1+k^2}{2})\left(\frac{2K^{\prime}(k)}{\pi}\right)^2={}_3 F_2\left(\begin{matrix}1/4& 3/4& 1/2\\1& 1\end{matrix};\frac{16k^2(1-k^2)^2}{(1+k^2)^4}\right)$$
with $\sqrt{2}-1\leq k\leq\sqrt{2}+1$, where $K$ and $K^\prime=K(\sqrt{1-k^2})$ are complete elliptic integrals of the first kind, $F$ is generalized hypergeometric function.
Question: Does A imply B(or B imply A)?