Physics question about a rope and a capstan

So I saw this question that my teacher sent to us and had some doubts. So there is a rope that is in contact with a cylinder (capstan) an angle theta, where theta is small. A slightly higher tension is applied at one end of the rope such that at that end there is T+deltaT. As the working out suggests, the other end has T. There is friction, which I assume acts perpendicular to the "half line" the line line that cuts the angle with which the rope is in contact with the cylinder in two. So, there will be a Normal reaction due to the capstan on the rope along the direction of that "half line" but on the opposite direction, so pointing away from the cylinder. Balancing forces gives me that N=2T*sin(theta/2) +DeltaT*sin(theta/2), so Friction force= mu*N, however the solution asserts that Fr=2*mu*T*sin(theta/2) only, has it used the approximation that the second term in my N is approximately 0?