?textplot

It is not quite clear what your maplet is expected to do. After typing: "1,2" in the Input Box, clicking the button "addition" results in the value 3.
If you want entering digits by pressing the number buttons, you must use at least different mames of variables to store the values. Etcetera. I'm affraid that a radical redesign of the program set-up is needed.

Perhaps you mean output in u_ξ-notation?

restart; with(PDEtools):
declare( );
du := diff(u(xi),xi,xi);

gives the desired output; and

OFF;
du

prints the second derivative of u in the normal layout.

I think from rounding errors, because you use floating point numbers in your input.

C := convert~(A,rational):
LinearAlgebra:-Determinant(C);

has only even powers of x in the output, as expected.

f := (-x^2+eta^2+2*x*xi+2*x*eta-xi^2-2*xi*eta-2*x*y-2*y*eta+2*xi*y+y^2) * 
     (-x^2-2*x*eta+eta^2+2*x*xi+2*xi*eta-xi^2+2*x*y-2*xi*y-2*y*eta+y^2) - 
     (x-xi)^4+6*((x-xi)^2)*(-y+eta)^2-(-y+eta)^4;
expand(f)

gives zero.

X := <seq(i,i=1..19)>:
Y := Vector( 19,
    RandomTools:-Generate( float(range=1..2)*10^integer(range=-13..-6), makeproc=true ) ):
plots:-logplot(X,Y, style=point);

Defining the function:

f := (x,y) -> x+(y-1)*arcsin(sqrt(x/y));

Second derivative wrt x:

D[1,1](f);

and with y=1 substituted:

D[1,1](f)(x,1);

You write for example

deltat*x^3

where problably is meant:

deltat(x)^3

or

deltat(x^3)

There are a lot more of this kind of errors/ambiguities.

What do you mean by "the wrong eigenfrequency"? What is the desired unit? degrees per second? What unit has the "2" in sin(2*t)? degrres per second indeed?

z := 3*exp(t*I):
plots:-complexplot( 6/z, t=-Pi..Pi );

or directly:

plot( [Re(6/z),Im(6/z), t=-Pi..Pi] );

No,

Nabla.Nabla(F)

is not the Hessian, but the Laplacian. You have to use parentheses to get the gradient.
To have full control over your input, you can better work in worksheet mode, and choose "Maple Notation"} for Input Display.

I suppose that you want the greatest of the two y-values in your list.
That is not always the first solution that is produced by the solve command.
So you may try

B := NULL:
for t from 10 by 2 to 100 do 
  h := [solve({x*y*z = 6*t^3, x-y-z = 0, x+y+z = 6*t}, {x, y, z})]:
  B := B,max(subs~(h,y)):
end do:
A := [B];

Do you mean a numerical approximation for the derivatives?
Are the θ[i] values equidistant, i.e. is θ[i+1] − θ[i] = h for all i?
Then you can use the central differentiation formula's

(I dont't understand your addendum)
If you really want to use Maple for this conversion:

> eq := A*sin(theta) + B*cos(theta) = C*cos(theta+delta);
> s := solve( identity(eq, theta), {A,B} );

Now you have A,B as functions of C and δ.

> solve( s, {C,delta} ): allvalues(%);

Your second question: you get a list of possibble equivalent trigonometric expressions by

> trigsubs(cos(theta+phi)*cos(theta+omega*t));

?copy

If A is a rtable, the assignment

B := A;

results in two names referring to the same structure.