Manipulate[Plot[{Sin[x],JacobiSN[x,m],Tanh[x]},{x,0,2*Pi},PlotLabel-> Style["sin(x) --> tanh(x)"]],{m,0,1,0.01}]

Shown above is a homotopy from sin(x) to tanh(x) via the Jacobi sn function. The Jacobi sn function has a parameter (we have denoted as m), and when m=0, the Jacobi sn function is the sine function and when m=1 the Jacobi sn function is the hyperbolic tangent function.

Manipulate[Plot[{Cos[x],JacobiCN[x,m],Sech[x]},{x,0,2*Pi},PlotLabel-> Style["cos(x) --> sech(x)"]],{m,0,1,0.01}]

Shown above is a homotopy from cos(x) to sech(x) via the Jacobi cn function. The Jacobi cn function has a parameter (we have denoted as m), and when m=0, 
the Jacobi cn function is the cosine function and when m=1 the Jacobi cn function is the hyperbolic secant function.

Manipulate[Plot[{1,JacobiDN[x,m],Sech[x]},{x,0,2*Pi},PlotLabel-> Style["1 --> sech(x)"]],{m,0,1,0.01}]

Shown above is a homotopy from the function identically equal to 1 to sech(x) via the Jacobi dn function. The Jacobi dn function has a parameter (we have denoted as m), and when m=0 the Jacobi dn function equals 1 identically and when m=1 the Jacobi dn function is the hyperbolic secant function.

TO EXPORT THESE TO A VIDEO FILE use this command: Export["FILENAMEYOUWANT.mov",X] and substitute the filename you want and for X use the entirety of one of the manipulate commands above.

Tom Cuchta 25 June 2012