What is the Optimal Traction Power for Unrolling Toilet Paper?

[haruspex]

It is easier to pull when the paper hang near to the wall.

because by the first equation:
Fz+Fg=Fr+T*cos(θ)

Yes.

we can cut out Fr and Fz:
Fg=T*cos(θ)

True, but it's the two values of Fz that you need to compare.

Is this the solution for my task?
I asked myself where ever the difference is between the two processes and it is just the point of action, is different.
I must not calculate the force acting when the paper is not on the wall, on the wall by the pulling force?

In the OP you wrote

Determine the force with which you have to pull the two Aufhängearten

which suggests you are actually supposed to determine the two values of Fz (as functions of the given variables). But since this is a translation it's hard for me to be sure.

 

[2013]

Yes, you've understood the task.

True, but it's the two values of Fz that you need to compare.

How can I compare the two different equations?
I'm finished or have I to do something?

 

[haruspex]

How can I compare the two different equations?

You have
(1) Fn = T*sin(θ)
(2) Fr = Fn*μ
(3) Fz = Fr
(4) sin(θ) = R/L
When hanging paper next to wall:
(5A) Fz+Fg = Fr+T*cos(θ)
When hanging paper away from wall:
(5B) Fz+Fg+Fr = T*cos(θ)
The given data are Fg, L and R, so for each case you want an equation involving only those and Fz. Try using (1)-(4) and (5A) to get such an equation.

 

[2013]

thank you so much for your help!