>>4
I get much more amusement from identities involving Jacobian elliptic functions like:
sn(x,k)^2+cn(x,k)^2=1
k^2*sn(x,k)^2+dn(x,k)^2=1
and
sn(x+y,k)=(sn(x,k)*cn(y,k)*dn(y,k)+sn(y,k)*cn(x,k)*dn(x,k))/(1-k^2*sn(x,k)^2*sn(y,k)^2)
cn(x+y,k)=(cn(x,k)*sn(y,k)-sn(x,k)*dn(x,k)*sn(y,k)*dn(y,k))/(1-k^2*sn(x,k)^2*sn(y,k)^2)
dn(x+y,k)=(dn(x,k)*dn(y,k)-k^2*sn(x,k)*cn(x,k)*sn(y,k)*cn(y,k))/(1-k^2*sn(x,k)^2*sn(y,k)^2)
These functions are rather useful and have direct applications in models of phenomena such as rocket motion, pendulum motion, and motion on spheres (Seiffert's spiral).
For instance, motion on the unit sphere in cylindrical coordinates (r, t, z) where the arc length of the path, s, has t=k*s with 0 < k < 1 has a trajectory of
(r, t, z) = (sn(s, k), k*s, cn(s, k)).
These functions are also extremely efficiently calculable via arithmetic-geometric mean (AGM) methods.