Step 1: Recall the definition of cosech.
$\text{cosech}\, x = \frac{1}{\sinh x}$. Given $\text{cosech}\, x = \frac{4}{5}$, so $\sinh x = \frac{5}{4}$.
Step 2: Recall the fundamental hyperbolic identity.
$\cosh^2 x - \sinh^2 x = 1$ (this is the hyperbolic analog of $\cos^2+\sin^2=1$).
Step 3: Solve for $\cosh^2 x$.
\[ \cosh^2 x = 1 + \sinh^2 x = 1 + \left(\frac{5}{4}\right)^2 = 1 + \frac{25}{16} = \frac{41}{16} \]
Step 4: Take the positive square root.
Since $\cosh x \geq 1 > 0$ for all real $x$: \[ \cosh x = \sqrt{\frac{41}{16}} \] This can also be written as $\frac{\sqrt{41}}{4}$, but as a fraction under the radical: $\sqrt{\frac{41}{16}}$.
Step 5: Match with the options.
Option 4 is $\sqrt{\frac{41}{16}}$. This matches exactly.
Step 6: State the answer.
\[ \boxed{\sqrt{\dfrac{41}{16}}} \]