Question

Let \(x:(0, \pi) \rightarrow \mathbf{R} ^2\) be the path given by \(x(t) = (\sin t, \cos t + \ln ( \tan\frac{t}{2} ) )\), where t is the angle of rotation of the vector x(t) from the negative y-axis. The image is called tractrix.
Show that the length of the segment of the tangent to the tractrix between the point of tangency and the y-axis is always equal to 1.