Thank you for providing the additional info and giving insight into the problem you are facing. Here are a couple Maple commands that will give you exactly what you are looking for!
s := [cos(t), sin(t)]; # define the position
v := diff(s, t); # take the derivative to get velocity
Nl := 30; # number of lines to draw
plots:-display([seq(plot([op(subs(t=2*Pi*(N/Nl), s+q*~v)), q=-1..1]), N=1..Nl)]);
The last line of Maple code is the idea I had alluded to earlier. Let me try to break it down a bit for you. This chunk of code will create a single plot of the tangent line when t is a given value and varies q from -1 to 1.
plot([op(subs(t=2*Pi*(N/Nl), s+q*~v)), q=-1..1])
Using the 'seq' command, we can create a list of multiple plot structures, each for a different time t using the index N. I made t vary between 0 and 2*Pi. Then using the plots package I displayed all the lines at once. Here is the result: