You have two sets of sets.  ?list for more details.  The attached worksheet manually converts each set into a list and then using commands from the ListTools package, arrives at your final desired result.  There are probably other, more elegant ways to do this.   

Download 221_set_list_sort.mws

While not a "proof", by inspection it looks equivalent.

>restart:

>eq:=4*cos(theta)^3-3*cos(theta);

>combine(eq);

Take a look at the help files for the select command.

?select

>restart:

>L:=[1,2,3,2,4,6,5,0,9,0,3,4,4];

>select(x->x<=3 and x>1,L);

>nops(%);

In one line:

>nops(select(x->x<=3 and x>1,L));

The syntax for the select command used above was taken from the following document:

www.mast.queensu.ca/~math418/m418oh/maple.pdf

I is the built in root of x^2 = -1, that is why you are getting an error.  You can change this using the line below at the beginning of your worksheet.  Then I[pk]:=10
will work. 

interface(imaginaryunit=j):

?interface for more help on this topic

The above code, using cut and then paste into v12 works fine.

restart:

plot(x^4 + 6*x^3 + 3*x^2 + 6,x=-6..1);

solve(x^4 + 6*x^3 + 3*x^2 + 6,x);

evalf(%);

This will give all solutions, including the complex ones.

fsolve(x^4 + 6*x^3 + 3*x^2 + 6,x=-6..-1);

This will return just the x values that result in y=0.  The range was chosen

after looking at the output of the plot command.  Look at the help files for

each of the solve commands (fsolve and solve) for more details. 

replace e with exp and look at the help pages for the solve command. Each of the commands below will bring up the help pages.  There are examples there that you can use.

?exp

?solve

I think this is what is meant:

ln(exp(1));

                 1

As Axel points above this, "e" does not have any particular meaning in Maple.

verify(sin(x)/cos(x),tan(x),'testeq');

works as well.

restart: with(plots):

implicitplot(x^2 + y^2 <= 4, x=-2..2, y=-2..2,filled=true,numpoints=1000);

For more help on this, ?implicitplot

The following link to the Maple Applications Center might be usefull to you.

?plottools[ellipse]  will get you started.

This site has a similar example that you can open in Maple and examine yourself.

Scroll 2/3 of the way down the page and open the file titled FITODES2.MWS

restart:

Q := x* 2500/(2+3*6^(1/2))^2/(3*6^(1/2)-2)^2;

simplify(%,[radical]);

I asked a similar question on comp.soft-sys.math.maple.  The square brackets "expand" the expression first, and then simplification occurs, w.r.t. radicals.  You can replace "radical" with "sqrt" and it will work as well.  This is not an explanation, but it will force a solution.  I have not seen any documentation on this.  If anybody has a link where this is documented, I would like to see it.

Correct.  That is the difference between it working and not working.  The rest is up to you.

 What happens if you change your last line to:

LineInt(VectorField(<x^2*y, x^3, x*y>), Path(<sin(t), cos(t), 1/4* (sin(t)^2 - cos(t)^2)>,t=0..2*Pi));

Do you get the answer that you are expecting?