For any $n\in R$, $\cosh \left(nx\right)+\sinh \left(nx\right)=$

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?

If $x\in \left(\frac{-\pi }{2},\frac{\pi }{2}\right),$ then $\mathrm{logsec}x=$