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"?

... to solve familiar problems using Maple. The attached exercises "test" and "test1" don't produce the familiar result. What am I doing wrong?

test.mw

test1.mw