@dharr The very same is accompliced with subs:

sqrt(1-cos(x)^2);

subs(cos(x)^2=1-sin(x)^2,%);

@nm No what I meant was simply that it gave a correct result if x is real, no more.

## Surely we have for x complex (including x real of course) that:

sqrt(1-cos(x)^2) = sqrt( sin(x)^2 )

## where both square roots are the the principle root.

So this relation holds in any case, but sqrt( sin(x)^2 ) is not the same as abs(sin(x)) for complex x, as e.g. x = 1+ I would show, since clearly abs(z) is real for any z.

This is as far as I get:

A:=sqrt(1-cos(x)^2);
B:=sqrt(sin(x)^2);
simplify(A) assuming x::complex;
%-B;
simplify(%); # 0

This works:

simplify(A) assuming x::real;
# Compare to:
plot(A,x=-Pi..Pi);

This returns NULL:

solve(sol,y) assuming y>1;

and so does this:

solve(sol,y) assuming y<1;

I can confirm that, but I went to Help/Check for Updates. 
Is the problem in fact that no update has been released?

If that is the case, then the response should have been "No update is availlable" or similar.

I filed an SCR.

@Rouben Rostamian  

In Windows 11 the scrollbar can be seen and works as expected also in Maple 2025.1.

I didn't get any hang. All reported "time expired".
`Standard Worksheet Interface, Maple 2025.1, Windows 11, June 12 2025 Build ID 1932578`

`Maple 2025.1, X86 64 WINDOWS, Jun 12 2025, Build ID 1932578`.

This message in the second group:

Error, (in expand) time expired

This message in the first group:

Error, (in simplify/getkernels) time expired

@sand15 On my very old other laptop I have some old versions.
It appears that Maple 2020.2 behaves as your Maple2015.2, i.e. gets 1. with Digits =10.
On my newer laptop I checked Maple 2021.2 and 2022.2. They get 1. too.
Maple 2023.2, 2024.2, and Maple 2025.1 get 0.

@vv You wrote:
"The strange fact is that any function of a complex number commutes (.) with any expression:
f(2+3*I).b - b.f(2+3*I) = 0. "

if f maps C into C then there is nothing strange about
f(2+3*I).b - b.f(2+3*I) = 0, whether or not b is a vector or a matrix.
The dot will be interpreted as multiplication by a scalar:

 

'(4+47*I).Vector([3,2])';
%;
'(4+47*I)*Vector([3,2])';
%;
'Vector([3,2]).(4+47*I)';
%;
'Vector([3,2])*(4+47*I)';
%;

It has always been lige that in Maple.
So the statement in the help page for dot:
If A and B are numbers (including complex and extended numerics such as infinity and undefined), then A . B = A*B.
ought to be modified to something like this:
If A or B is a number (including complex and extended numerics such as infinity and undefined), then Then the dot in A.B and B.A will be regarded as multiplication by a scalar

The help page continues with this:
 

If one of A and B is a Matrix or a Vector, and the other is a Matrix, Vector or constant and the previous case does not apply, then their product is computed by using LinearAlgebra[Multiply].  See also simplify/rtable. 

Here the case when one of A or B is a constant is covered by the the documentation for
LinearAlgebra[Multiply], but the present simple case drowns in all the cases that are handled.

@vv Here is a counterexample:
 

#f(2+3*I).b - b.f(2+3*I);
restart;
f:=x->x*Matrix([[1,2],[3,4]]);
b:=Matrix([[5,6],[7,8]]);
f(2+3*I).b - b.f(2+3*I);

The answer from the last line is:
Matrix([[-8 - 12*I, -24 - 36*I], [24 + 36*I, 8 + 12*I]]);

This is at least as old as Maple 8; I just checked.

The description in ?dot is basically the same as in Maple 2025.
The second line for example in Maple 2025 is

If A and B are numbers (including complex and extended numerics such as infinity and undefined), then A . B = A*B.

This is exactly the same as in Maple 8.

The behavior is indeed strange (at least somewhat).

@vv Here is an assuming version that works:

restart;

puzzle := sum(cos(k*Pi/(2*n + 1))^4, k = 1 .. n);

expand(convert(puzzle,exp));
simplify(%) assuming n::posint;

@vv This version works:

restart;
puzzle := sum(cos(k*Pi/(2*n + 1))^4, k = 1 .. n);
simplify(expand(convert(puzzle, exp)),assume=n::posint);

See simplify/details.

@janhardo I know how to make a screenshot.
I usethe special  PrtSc.

What I want removed is this whole thing:
 

@vv Thank you for the reference. Now i have Maple 2025.1.
I had tried Check for Updates in Maple 2025.0 a couple of times, but the answer was "No updates available".