Thinking about some physical problem, I came across the following identity: $$\phi^2\Pi\left(-\phi,\frac{1}{\sqrt{2}}\right)+\phi^{-2}\Pi\left(\phi^{-1},\frac{1}{\sqrt{2}}\right)=\frac{\pi}{\sqrt{2}}+K\left(\frac{1}{\sqrt{2}}\right )=\frac{\pi}{\sqrt{2}}+\frac{8\pi\sqrt{\pi}}{\Gamma\left(-\frac{1}{4}\right)^2},$$ where $\phi=\frac{\sqrt{5}+1}{2}$ is the golden ratio, and $$K(k)=\int\limits_0^{\pi/2}\frac{d\theta}{\sqrt{1-k^2\sin^2{\theta}}},\quad \Pi(n,k)=\int\limits_0^{\pi/2}\frac{d\theta}{(1-n\sin^2{\theta})\sqrt{1-k^2\sin^2{\theta}}}$$ are complete elliptic integrals of the first and third kind, respectively. Is this identity known?
In fact, more general identity is $$\frac{\alpha}{\alpha-1}\Pi\left(-\frac{1}{\alpha-1},k\right)+ \frac{\alpha-1}{\alpha}\Pi\left(\frac{1}{\alpha},k\right)=\frac{\pi}{2\sqrt{1-k^2}}+K(k), \tag 1$$ where $\alpha>1$, and $$k^2=\frac{1}{1+\alpha(\alpha-1)}.$$
P.S. As მამუკა ჯიბლაძე noted (and I have verified it), these are particular cases of 19.7.iii in DLMF. However I obtained them from an even more general identity: $$\frac{\alpha+\beta+1}{2\beta}\Pi\left(-\frac{\alpha-\beta}{\beta},k\right)+ \frac{\alpha+\beta-1}{2\alpha}\Pi\left(\frac{\alpha-\beta}{\alpha},k\right)=\\ \frac{1-\alpha+\beta}{1+\alpha-\beta}\Pi(n,k)+K(k), \tag 2$$ where $\alpha\ge 0$, $\beta\ge 0$, $\alpha\ge \beta$, and $$n=\frac{4(\alpha-\beta)}{(1+\alpha-\beta)^2},\quad k^2=\frac{4(\alpha-\beta)}{(1+\alpha-\beta)^2+4\alpha\beta},$$ in the limit $\alpha-\beta\to 1_-$. Where can I find proof of 19.7.iii in DLMF?
P.P.S. It seems (1) and (2) are valid under more general conditions than indicated. For example, from (2), when $\alpha=-\beta=1/12$, we obtain $$12\Pi\left(2,\frac{1}{\sqrt{2}}\right)+\frac{5}{7}\Pi\left(\frac{24}{49}, \frac{1}{\sqrt{2}}\right)=-K\left(\frac{1}{\sqrt{2}}\right)= - \frac{8\pi\sqrt{\pi}}{\Gamma\left(-\frac{1}{4}\right)^2},$$ and from (1) then $\alpha=1/2$ we obtain (up to an ambiguity in choosing sign in $\sqrt{-1}=\pm i$) $$2\Pi\left(2,\frac{2}{\sqrt{3}}\right )+K\left (\frac{2}{\sqrt{3}}\right )= -i\frac{\sqrt{3}}{2}\pi.$$ At last, when $\alpha=-\beta=-1/12$, (2) gives $$12\Pi\left (2,i\right)-\frac{7}{5}\Pi\left (-\frac{24}{25},i\right)=K(i)=\frac{\Gamma\left(\frac{1}{4}\right)^2}{4\sqrt{2}\pi}.$$ Wolfram Alpha indicates that all identities are correct.
