I have to solve the next problem for a course at university:
Parametrise the grim's reaper curve $y = -\ln(\cos x)$ with $x \in \bigl(-\frac{\pi}{2}, \frac{\pi}{2}\bigr)$ by the arc length using the parametric equation of itself and the fundamental theorem of the local theory of curves.
So I have proceeded to write the equation in a parametric way: $$ \begin{cases} x = t \\ y = -\ln(\cos t) \end{cases} $$
And I have got the arc length integrating from $0$ to $T$: $$ S = \ln(\sec t + \tan t) $$
Now I think I have isolate the $T$ and substitute in the $x,y$ equations, but I think it is not possible? Maybe have I missed an step?
Thanks in advance