how can we prove that length of arc in a circle is equal to radius of circle multiplied by angle between extreme radii or sector?
for a central angle in a circle, its radian angle measure, , is defined as the ratio of the length of the intercepted arc, , divided by the length of the radius, ...
using that definition, you should be able to arrive at the desired result rather easily.
how can we prove that length of arc in a circle is equal to radius of circle multiplied by angle between extreme radii or sector?
It is really a matter of definition as opposed to proof.
One radian is defined to be the measure of a central angle in any circle that subtends an arc of the circle equal in length to the radius of the circle.