Answer
$\tanh x=\dfrac{-3}{5}$
$sech x=\dfrac{4}{5}$;
$csch x=\dfrac{4}{-3}$;
$coth x=\dfrac{5}{-3}$
Work Step by Step
As we are given that $\sinh x=\dfrac{-3}{4}$
Need to find all the hyperbolic functions.
Use identity: $\cosh^2 x-\sinh^2x=1$
or, $\cosh^2 x-(\dfrac{-3}{4})^2=1 \implies \cosh^2 x=\dfrac{25}{16}$ and $\cosh x= \dfrac{5}{4}$
Now, $\tanh x= \dfrac{\sin hx}{\cosh x}=\dfrac{\dfrac{-3}{4}}{\dfrac{5}{4}}=\dfrac{-3}{5}$
$sech x= \dfrac{1}{\cosh x}=\dfrac{1}{\dfrac{5}{4}}=\dfrac{4}{5}$;
$csch x= \dfrac{1}{\sinh x}=\dfrac{1}{\dfrac{-3}{4}}=\dfrac{4}{-3}$;
$coth x= \dfrac{1}{\tanh x}=\dfrac{1}{\dfrac{-3}{5}}=\dfrac{5}{-3}$
