Problem

(a) Starting from the definitions of $\tanh$ and $\operatorname{sech}$ in terms of exponentials, prove that

$$\tanh^2 t + \operatorname{sech}^2 t = 1.$$

(b) The curve $C$ has parametric equations

$$\begin{align*} x =&\, \ln(\cosh t), \quad y = \tan^{-1}(\sinh t), \quad \text{for } 0 \le t \le 1. \end{align*}$$

Find the length of $C$.