When calculating limits of real-valued functions, sometimes (especially in competitions) tricky approaches are taken using pen and paper. I repeatedly encountered the simple conclusion that, for example, for natural k, the value sin(k*pi) = 0. Thus, the function value is determined logically without specifying a specific number. There are numerous other examples of this that can easily be constructed.
My question after unsuccessful attempts using "assume" is:
How, for example, does Maple determine the value of sin(k*pi) from the assumption "k is natural" alone? Are such prominent values ​​implemented in tables?

In the attached file, I want to restrict the indices of the summation to gcd(m,n)=1. How does this work?

test.mw

In the attached file, I was unable to calculate the limit values ​​L and M. Please help me.

test.mw

In the attached file, I'm trying to calculate a limit. After a long calculation, I've given up. I'm asking for advice on how to perform the calculation effectively in Maple. I know the solution using the pen and paper method (pi/4+1/2*ln(2)).

test.mw

I recently solved the following Diophantine equations:
tan(3*pi/x)+4*sin(2*pi/x)-sqrt(x)=0
and
tan(13*pi/x)+4*sin(19*pi/x)-sqrt(x)=0
Unlike my old "pencil and paper" solution, I used Maple to practice with some sub-calculations to get some guesses for the solution. To confirm my guesses, I inserted them into the equations and used "simplify." The result was "zero." Is this "zero" the mathematically exact zero, or does Maple display a very small real number as "zero" after applying "simplify"?