Download Check.mw

f2:=cos(y-Pi/3):
is(expand(f1/f2)=1);

                         true

Formally, the expressions  f1  and  f2  do not coincide for all values x and y. For example we get an undefined expression  0/0  for x=0, y=0 :
 

eval([numer(f1),denom(f1)],[x=0,y=0]);

                         [0, 0]
 

m := n->piecewise(type(n,even),0, 1):
f := proc(a,b)
if type(a*b,integer) then return m(a*b) else FAIL fi;
end:
 
f(a,b);
f(2,3);
f(1,3);
f(a,3);

sin(30*Pi/180);
30^`°`;

Example:

restart;
CurveFitting:-LeastSquares([[0, 1], [1, 2], [2, 3], [3, 10]], x);
A, B:=coeffs(%);

                               1/5+14*x*(1/5)
                             A, B := -1/5, 14/5

Good question - vote up.

restart;
P:=proc(n::nonnegint) 
local x, x0, v0, q, v:= <<x>>, s:= <<0>>, p:= <<1>>; 
   for x0 to n do v0:=eval(v,x=x0); p:= eval(p.v0,x=x0); s:= eval(s+v0, x=x0) od; 
   p, s; 
end proc:

P(6);

I have never used the command  rtable_eval  and do not know the reasons for this error.  A workaround below:

restart;
P:=proc()
local v;
v:= <x>;  
eval(v, x=1);
end proc():
P();

`&Delta;a`;
                                    

Should be  u*v  instead of  uv .

diff(BesselJ(alpha, sqrt(u^2+v^2-2*u*v*cos(phi))), u);

Here are 2 easy ways to do this without a long fraction bar:

restart;
(1/x)*(x^3-2*x^2+5*x-7);  # With long fraction bar
``(1/x)*(x^3-2*x^2+5*x-7);  # The first way
(x)^``(-1)*(x^3-2*x^2+5*x-7);  # The second way

Here are all the $ 200 shopping options, sorted by increasing Gb:

sort([seq(seq(seq([s=32*x+64*y+128*z,['x'=x,'y'=y,'z'=z]],z=floor((200-15*x-20*y)/30)),y=0..floor((200-15*x)/20)),x=0..floor(200/15))], key=(p->rhs(p[1]))):
select(p->floor((200-eval(15*x + 20*y + 30*z,p[2]))/15)<1 and floor((200-eval(15*x + 20*y + 30*z,p[2]))/20)<1 and floor((200-eval(15*x + 20*y + 30*z,p[2]))/30)<1, %)[];
nops([%]); # The number of all the possibilities

       [s = 416, [x = 13, y = 0, z = 0]], [s = 448, [x = 10, y = 2, z = 0]], [s = 448, [x = 12, y = 1, z = 0]], [s = 480, [x = 9, y = 3, z = 0]], [s = 480, [x = 11, y = 0, z = 1]], [s = 512, [x = 6, y = 5, z = 0]], [s = 512, [x = 8, y = 2, z = 1]], [s = 512, [x = 8, y = 4, z = 0]], [s = 512, [x = 10, y = 1, z = 1]], [s = 544, [x = 5, y = 6, z = 0]], [s = 544, [x = 7, y = 3, z = 1]], [s = 544, [x = 9, y = 0, z = 2]], [s = 576, [x = 2, y = 8, z = 0]], [s = 576, [x = 4, y = 5, z = 1]], [s = 576, [x = 4, y = 7, z = 0]], [s = 576, [x = 6, y = 2, z = 2]], [s = 576, [x = 6, y = 4, z = 1]], [s = 576, [x = 8, y = 1, z = 2]], [s = 608, [x = 1, y = 9, z = 0]], [s = 608, [x = 3, y = 6, z = 1]], [s = 608, [x = 5, y = 3, z = 2]], [s = 608, [x = 7, y = 0, z = 3]], [s = 640, [x = 0, y = 8, z = 1]], [s = 640, [x = 0, y = 10, z = 0]], [s = 640, [x = 2, y = 5, z = 2]], [s = 640, [x = 2, y = 7, z = 1]], [s = 640, [x = 4, y = 2, z = 3]], [s = 640, [x = 4, y = 4, z = 2]], [s = 640, [x = 6, y = 1, z = 3]], [s = 672, [x = 1, y = 6, z = 2]], [s = 672, [x = 3, y = 3, z = 3]], [s = 672, [x = 5, y = 0, z = 4]], [s = 704, [x = 0, y = 5, z = 3]], [s = 704, [x = 0, y = 7, z = 2]], [s = 704, [x = 2, y = 2, z = 4]], [s = 704, [x = 2, y = 4, z = 3]], [s = 704, [x = 4, y = 1, z = 4]], [s = 736, [x = 1, y = 3, z = 4]], [s = 736, [x = 3, y = 0, z = 5]], [s = 768, [x = 0, y = 2, z = 5]], [s = 768, [x = 0, y = 4, z = 4]], [s = 768, [x = 2, y = 1, z = 5]], [s = 800, [x = 1, y = 0, z = 6]], [s = 832, [x = 0, y = 1, z = 6]]
                                                                            44

We see that  s=832  can only be obtained in one way.

Edit.
 

[seq(seq(`if`(5*x + 3*y = 100, [x,y], NULL), x=0..20), y=0..20)];
plots:-pointplot(%);

Here are the Euclidean algorithm and the extended Euclidean algorithm.

Download Euclid.mw

You can use the commands  plots:-arrow, plots:-display, LinearAlgebra:-CrossProduct  to plot vectors, the results of operations on them, and animations of these. See examples below:

Download Plotting_Vectors.mw

Try the  Explore  command for this:

It can be noted that changing the parameter  C  has very little effect on the graph

Download 2param.mw