The question is in regards to the two lemniscatic elliptic functions, often called the 'sine lemniscate' and 'cosine lemniscate' functions. I have been trying to prove the following identity:
\begin{align} \frac{sl(x) sl'(y) + sl'(x) sl(y)}{1 + sl(x)^2 sl(y)^2} \end{align}
as equal to: \begin{align} \frac{sl(x) cl(y) + sl(y) cl(x)}{1 - sl(x)cl(y)sl(y)cl(x)} \end{align}
I have had no success. I've attempted using the following relations:
$$ (1+sl(x))(1+cl(x)) = 2 $$ $$ sl(x)^2 + cl(x)^2 + sl(x)^2cl(x)^2 = 1 $$ $$ sl(x) = \sqrt{\frac{1 - cl(x)^2}{1 + cl(x)^2}} $$ $$ sl(\frac{\varpi}{2} - x) = cl(x) $$ and while I've managed to mix things up nicely, I've not successfully gone from one identity to the other.
This has been vexing me for a bit. Can anyone please help?