It is not difficult to manually check the validity of the identity  sqrt(x+2*sqrt(x-1)) = sqrt(x-1)+1 , which is true at least for all  x>=1 . I don't know of any Maple command that directly makes this simplification:

expr:=sqrt(x+2*sqrt(x-1));
simplify(expr);
simplify(expr) assuming x>=1;

It was possible to simplify it only by making a variable substitution  x=y+1 (and then a reverse substitution  y=x-1 ) :

expr1:=subs(x=y+1,expr);
simplify(expr1);
subs(y=x-1, %);

By the way,  the CAS Mathematica also cannot cope with simplifying the expression  expr .

is(cos(x)=sqrt(1-sin(x)^2) or cos(x)=-sqrt(1-sin(x)^2)) assuming real;

                                          false  

In a simpler situation the result is correct

is(x=0 or x<>0) assuming real;

                                           true         

Here's a problem that I think works well with Maple.
Three runners, A, B, and C, are running on parallel tracks (each running at a constant speed, but not necessarily at the same speed as the others). At the start, the area of ​​triangle ABC is  2 , and after 5 seconds it is  3 . What might this area be after another  5  seconds?

Here's a puzzle for geometry lovers. It has a very short manual solution, but it's not that easy to find. Of course, you can solve it in Maple using coordinates. You need to find the radius of these two identical circles.

is(abs(x)=max(x,-x)) assuming real;

#  FAIL

I wonder if this will work in newer versions of Maple?