@yangtheary All answers in the last lines of code. I have added the required notation for clarity.

@Yo  You must provide a list of the main variables:

degree(alpha*x[1]^3+beta*x[1]*x[2]^2, [x[1],x[2]]);

                                               3

In Maple 2018.2 on Windows 10, there is no such problem. All settings are the same as yours:

@lcz  See my upgraded answer.

@lucaud

restart;
eq := cos(2*x)-sin(x)*(sin(x)^2+1)^(1/2)+cos(x)^2*sin(x)/(sin(x)^2+1)^(1/2):
subs([cos(2*x)=1-2*sin(x)^2, cos(x)^2=1-sin(x)^2], eq);

Or more automatically

restart;
eq := cos(2*x)-sin(x)*(sin(x)^2+1)^(1/2)+cos(x)^2*sin(x)/(sin(x)^2+1)^(1/2):
expand(subs(x=arcsin(y), eq));
subs(y=sin(x), %);

@OscarSteenstrup  These are simply parametric equations of a conical surface, where polar coordinates on the yOz plane are taken as parameters.
If the equation of a curve on a plane  xOy  in Cartesian coordinates is  f(x,y)=0  and this curve rotates around the  Ox-axis, then the equation of a surface of revolution is  f(x, sqrt(y^2+z^2))=0 . Your cone can also be defined by this equation. The point is that if the surface is specified parametrically, then the quality of the drawing is usually much higher. See

f:=y=x/2;
plots:-implicitplot3d(eval(f, y=sqrt(y^2+z^2)), x=0..12, y=-5..5, z=-5..5);

@Scot Gould  You missed  the theta variable before the tangent.

@Johan159  Should be

restart;
f:=(x,y)->2*x^2+5*y^3+5:
plot(x->f(x,5), 0..10, labels=[x,z]); # The curve in 2D
plot3d(f, 0..10, 5..5, axes=normal, labels=[x,y,z]); # The same curve in 3D

@Preben Alsholm  Thank you.  In general case, you are right, but in this example there is no error, because

series(cos(x),x=0, 6) = series(cos(x),x=0, 5);
 

@Qruze  This is strange. I do not have Maple 2020 and cannot test this behavior. But this code in Maple versions 2015 - 2018 solves the example without any problems.

This issue has been discussed in detail in this thread  https://www.mapleprimes.com/questions/231640-Writing-A-Procedure-

@AHSAN  You have to do something yourself. For other questions, see the help on the  commands LinearAlgebra:-Pivot  and  LinearAlgebra:-RowOperation . 

@nm  I don't know the answer to the last question. I've never used parentheses when working with matrices.

@oggsait  Use the  legend  option for this.

@ik74  In your drawing, the center of the circle should be to the right and not to the left of the centerline.
The code for this case:

restart;
a:=2: R:=-8: z:=sqrt(R^2-(r-a+R)^2):
plot3d([[r,phi,z],[r,phi,-z]], phi=0..2*Pi, r=2..3, coords=cylindrical, scaling=constrained);