Draw the graph of $\coth x$. Find the derivative of $\coth x$.

If $x={\log }_e\left[\cot \left(\frac{\pi }{4}+\theta \right)\right]$ and $\theta \in \left(\frac{-\pi }{4},\frac{\pi }{4}\right),$ then consider the following statements:

$\mathrm{I}: \cosh x=\sec 2\theta$
$\mathrm{II}:$ $\sinh x=-\tan 2\theta$

Then which one of the following options is true?

Sketch the graph of $y=\left|3\cosh x-5\right|$.

Sketch the graph of $y=4-cosech\left(x-2\right)$.