this may help you
he functión Cot[x/2] is not continuos in the points x=2nπ, where n=0,1,2,3,... You can check it with a calculator. So the function is not continuos in the domain the problem gives you, so the Rolle's theorem can not be applied. If the inteverval was, [π/2,3π/2] Then we could apply the Rolle's theorem.