@Carl Love  Your plot  does not prove anything, because it is built on a finite set of points, and the break point is somewhere between them. Here is another plot  on a very small interval, inside which there is a break point (the zero of the denominator):

Digits:=20:
fsolve(denom(x(t)));
plot(x(t), t=-0.00077376199177721..-0.00077376199177719, -5000..5000);

@666 jvbasha 

Download ContoursWithLabels11.mw

@radaar  For example, when you define a procedure using  arrow-notation:

x:=10: t:=20:
f:=x->x^2:
g:=t->t^2:
f(3); g(3);

f  and  g  are  the same function.

The solution to similar and much more complex examples is automated in this post
https://www.mapleprimes.com/posts/145922-Perimeter-Area-And-Visualization-Of-A-Plane-Figure-

@SanzharMukatay  This is correct, but a little cumbersome. For example, if you set the law of motion in a list, the components of which represent the corresponding coordinates, then you can immediately differentiate this list and get speed components in the form of a list too. The same goes for acceleration:

xy:=[3*t,4*t^2+1]:
v_xy:=diff(XY,t);
a_xy:=diff(v_xy,t);

                                               v_xy := [3, 8 t]
                                                a_xy := [0, 8]

@Axel Vogt  Probably, vv deliberately swapped the numerator with the denominator in order to obtain the desired result for OP. I actually did the same in my answer.

@AHSAN  You can use the same data, only each sublist needs to be reversed:

Download help_new1.mw

@Rouben Rostamian  Thank you. I dont know. The case  assuming integer is more complicated because we must consider the case separately when one parameter is  0 .

int(cos(m*t)*cos(0*t), t=-Pi..Pi, allsolutions) assuming integer;
convert(%, piecewise, m);

@Grigoriy Yashin  I added 1 short line:

Download trigon_new.mw

@Tyttemus  You yourself see that on the right side of the equality  k = ...  the variable  V  is not reduced, that is,  k  depends on  V . Therefore, it is impossible to remove  V .

Here is a very simple example illustrating your problem. Let you have the equation  3*x+2*y=0 . Then it is easy to find  x/y = -2/3. But if we slightly change the equation  3*x+2*y+1=0 , then it is impossible to find  x/y , that it does not depend on either  x  or  y .

@Carl Love  Thank you for this improvement.

@Carl Love  This probably only works in recent versions of Maple.

In Maple 2018.2 :

[for i from 11 to 20 do i^2 od];

     Error, reserved word `for` unexpected

@nm   Try  color="DarkSlateGray"  instead of color=gray .

You should present all of this in text form using Maple syntax. Why should someone do this for you?

@Teep  Of course, all this is easy to do on one plane:

restart; with(plots): 
z := (x, y)->(x^2+y*x+y^2)*sqrt((1-y^2)^2/(x+y)^2-(x-y)^2): 
Colors := [yellow, orange, red, green, blue, khaki, violet, cyan,pink,magenta]: 
plot([seq([x, z(x, y), x = 0 .. 1], y = 0 .. 0.9, 0.1)], color = Colors, thickness = 2, legend = [seq(y = C, C = 0 .. 0.9, 0.1)], labels = [x, z]);

Edit.