Definition of Inverse Hyperbolic Functions

If the range of $\sec h^{-1}x+\mathrm{cosec}{h}^{-1}x$ is $[a, b)$, then

Prove that ${sech}^{-1}x\equiv \ln \left(\frac{1+\sqrt{1-x^2}}{x}\right).$

Prove that ${\tanh }^{-1}x\equiv \frac{1}{2}\ln \left(\frac{1+x}{1-x}\right)$

Prove that ${\cosh }^{-1}x\equiv \ln \left(x+\sqrt{x^2-1}\right).$

Find the exact value of: ${\cosh }^{-1}\frac{5}{4}$

Find the exact value of: ${\tanh }^{-1}\frac{2}{3}$

Find the exact value of: ${\sinh }^{-1}3$

Prove that ${sech}^{-1}x\equiv \ln \left(\frac{1+\sqrt{1-x^2}}{x}\right).$

Prove that ${\tanh }^{-1}x\equiv \frac{1}{2}\ln \left(\frac{1+x}{1-x}\right)$

Prove that ${\cosh }^{-1}x\equiv \ln \left(x+\sqrt{x^2-1}\right).$

Find the exact value of: ${\cosh }^{-1}\frac{5}{4}$

Find the exact value of: ${\tanh }^{-1}\frac{2}{3}$

Find the exact value of: ${\sinh }^{-1}3$

If for $\sinh x=3$, the value of $x$ obtained is in the form of ${\log }_e(a+\sqrt{b})$, then find the value of $a+b$.

$e^{\left(\sec h^{-1}\frac{1}{2}+\tan h^{-1}\frac{1}{2}+\sin h^{-1}\frac{1}{2}\right)}=$

If $f\left(x\right)=\cos h^{-1}\left(\frac{1-x}{1+x}\right)$ is well defined, then $f^{\text{'}}\left(x\right)=$

${coth}^{-1}3+{\tanh }^{-1}\frac{1}{3}-{cosech}^{-1}\left(-\sqrt{3}\right)=$

If $\sinh x=\frac{3}{4}$ and $\cosh y=\frac{5}{3}$ then $x+y=$

If $\tan h^{-1}x=alog\left(\frac{1+x}{1-x}\right),\left|x\right|<1,$ then a is equal to

$x=\log \left(\frac{1}{y}+\sqrt{1+\frac{1}{y^2}}\right)\Rightarrow y$ is equal to