I need some help with question 4 in section 1.3 in Baby Do-Carmo textbook in DG.
The question asks: Let $\alpha(t):(0,\pi)\rightarrow R^2 $ be given by: $ \alpha(t)= (\cos(t), \cos(t) +\log(\tan(t/2)) $ its image is called the tractrix.
Question b, asks to prove that the length of the segment of the tangent of the tractrix between the point of tangency and the y axis is constantly 1.
Now the angle between $\alpha$ and the y axis is t.
So basically if I were to use the sine theorem from trig, where \frac{S}{\sin(t)} = \frac{|\alpha(t)|}{\sin(\pi-(t+\angle \alpha(t) \alpha '(t)))} Where S is the required line segment I am looking for.
Now I am only left with calculating the angle between $\alpha(t)$ and \alpha '(t), is this about right, or I am way off here?
It's hell of a calculation if I am right (and it's really rare when I am). Thanks.