Jacobi defined the elliptic function by
$u=\int_0^{\phi}\frac{d\theta}{\sqrt{1-k^2\sin^2(\theta)}}$ ,$am(u)=\phi,sn(u)=\sin(am(u)),cn(u)=\cos(am(u))$
$dn(u)=\sqrt{1-k^2sn^2(u)},k'^2+k^2=1$
and he set $K=\int_0^{\pi/2}\frac{d\theta}{\sqrt{1-k^2\sin^2(\theta)}},K'=\int_0^{\pi/2}\frac{d\theta}{\sqrt{1-k'^2\sin^2(\theta)}}$
then he listed without proof these five formulas:
$\sin(am(K-u))=\frac{cn(u)}{dn(u)},\cos(am(K-u))=\frac{k'sn(u)}{dn(u)},\frac{d(am(K-u))}{du}=\frac{k'}{dn(u)}$
$\tan(am(K-u))=1/(k'\tan(am(u)),\cot(am(K-u))=k'/\cot(am(u))$
I tried a lot but can't figure out how these are deduced? I used the addition formula but it didn't work. Can anyone give a detailed deduction about these formulas?