# Classification of the surface of revolution with -1 constant Gaussian curvature

I know that under the arc length parametrization for the curve $$t\mapsto (x(t),0,z(t))$$, the Gaussian curvature of the surface of revolution is $$K = -\frac{x''}{x}$$. So, $$K = -1\iff x'' = x$$. Now, I know that the pseudosphere $$\gamma(t) = (sin(t),0,cos(t)+ln(tan(t/2)))$$ for $$t\in (\pi/2,\pi)$$ has -1 Gaussian curvature. As the arc length of $$\gamma(t)$$ is $$Arclength(\gamma) =cot(s)$$, $$\gamma(cot^{-1}(s))$$ is an arc length parameterization of it. But, $$\frac{d^2}{ds^2}sin(cot^{-1}(s))\neq sin(cot^{-1}(s))$$. What's wrong in here?

First, the arclength $$s$$ satisfies $$ds/dt =-\cot t$$. You lost a sign and an integral. So, in fact, $$s=-\log\sin t$$. Now finish correctly.