Forward-reverse trigonometric formulae

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A "forward-reverse" trigonometric function is a function of the form foo(arcbar(x)) where foo(x) and bar(x) are trigonometric functions (one of sine, cosine, tangent, cosecant, secant, cotangent).

arcbar(x) (or equivalently bar^-1(x)) is the inverse function of bar(x). So foo(arcbar(x)) is foo of the inverse of bar, hence "forward-reverse".

The tables below list these identities for every combination of trigonometric formulae for both circular and hyperbolic trigonometric functions. Derivations of these formulae are detailed in hidden sections in order to remove clutter, but you can expand these hidden section to read them by clicking on the "show" tags.

Circular functions

Forward-reverse circular trigonometric formulae
foo(arcbar(x))	arcsin	arccos	arctan	arccosec	arcsec	arccot
sin	$x$	$\sqrt{1-x^2}$	$x/\sqrt{1+x^2}$	$1/x$	$\sqrt{x^2-1}/x$	$1/\sqrt{1+x^2}$
cos	$\sqrt{1-x^2}$	$x$	$1/\sqrt{1+x^2}$	$\sqrt{x^2-1}/x$	$1/x$	$x/\sqrt{1+x^2}$
tan	$x/\sqrt{1-x^2}$	$\sqrt{1-x^2}/x$	$x$	1 $/\sqrt{x^2-1}$	$\sqrt{x^2-1}$	$1/x$
cosec	$1/x$	$1/\sqrt{1-x^2}$	$\sqrt{1+x^2}/x$	$x$	$x/\sqrt{x^2-1}$	$\sqrt{1+x^2}$
sec	$1/\sqrt{1-x^2}$	$1/x$	$\sqrt{1+x^2}$	$x/\sqrt{x^2-1}$	$x$	$\sqrt{1+x^2}/x$
cot	$\sqrt{1-x^2}/x$	$x/\sqrt{1-x^2}$	$1/x$	$\sqrt{x^2-1}$	$1/\sqrt{x^2-1}$	$x$

Proofs for the above formulae

The proofs for the circular trigonometric formulae use the following identities for the functions:

$\sin(\theta)=O/H\\\cos(\theta)=A/H\\\tan(\theta)=\sin(\theta)/\cos(\theta)=O/A\\\text{cosec}(\theta)=1/\sin(\theta)=H/O\\\sec(\theta)=1/\cos(\theta)=H/A\\\cot(\theta)=1/\tan(\theta)=A/O$

When $0<\theta<\pi/2$ is an angle in a right-angled triangle, O represents the length of the triangle's side opposite to the angle, A represents the length of the triangle's non-hypotenuse side adjacent to the angle, and H represents the length of the triangle's hypotenuse. They are related by the conversion identity:

$O^2+A^2=H^2$

sin(arccos(x))
$\begin{eqnarray}\sin\left(\arccos(x)\right)&=&\frac{O}{H}\\&=&\frac{\sqrt{H^2-A^2}}{H}\\&=&\sqrt{1-\left(\frac{A}{H}\right)^2}\\&=&\sqrt{1-\left[\cos\left(\arccos(x)\right)\right]^2}\\&=&\sqrt{1-x^2}\end{eqnarray}$

sin(arctan(x))
$\begin{eqnarray}\sin\left(\arctan(x)\right)&=&\frac{O}{H}\\&=&\frac{O}{\sqrt{A^2+O^2}}\\&=&\frac{\left(\frac{O}{A}\right)}{\sqrt{1+\left(\frac{O}{A}\right)^2}}\\&=&\frac{\tan\left(\arctan(x)\right)}{\sqrt{1+\left[\tan\left(\arctan(x)\right)\right]^2}}\\&=&\frac{x}{\sqrt{1+x^2}}\end{eqnarray}$

sin(arccosec(x))
$\begin{eqnarray}\sin\left(\text{arccosec}(x)\right)&=&\frac{O}{H}\\&=&\frac{1}{\left(\frac{H}{O}\right)}\\&=&\frac{1}{\text{cosec}\left(\text{arccosec}(x)\right)}\\&=&\frac{1}{x}\end{eqnarray}$

sin(arcsec(x))
$\begin{eqnarray}\sin\left(\text{arcsec}(x)\right)&=&\frac{O}{H}\\&=&\frac{\sqrt{H^2-A^2}}{H}\\&=&\frac{\sqrt{\left(\frac{H}{A}\right)^2-1}}{\left(\frac{H}{A}\right)}\\&=&\frac{\sqrt{\left[\sec\left(\text{arcsec}(x)\right)\right]^2-1}}{\sec\left(\text{arcsec}(x)\right)}\\&=&\frac{\sqrt{x^2-1}}{x}\end{eqnarray}$

sin(arccot(x))
$\begin{eqnarray}\sin\left(\text{arccot}(x)\right)&=&\frac{O}{H}\\&=&\frac{O}{\sqrt{O^2+A^2}}\\&=&\frac{1}{\sqrt{1+\left(\frac{A}{O}\right)^2}}\\&=&\frac{1}{\sqrt{1+\left[\cot\left(\text{arccot}(x)\right)\right]^2}}\\&=&\frac{1}{\sqrt{1+x^2}}\end{eqnarray}$

cos(arcsin(x))
$\begin{eqnarray}\cos\left(\arcsin(x)\right)&=&\frac{A}{H}\\&=&\frac{\sqrt{H^2-O^2}}{H}\\&=&\sqrt{1-\left(\frac{O}{H}\right)^2}\\&=&\frac{\sqrt{H^2-O^2}}{H}\\&=&\sqrt{1-\left[\sin\left(\arcsin(x)\right)\right]^2}\\&=&\sqrt{1-x^2}\end{eqnarray}$

cos(arctan(x))
$\begin{eqnarray}\cos\left(\arctan(x)\right)&=&\frac{A}{H}\\&=&\frac{A}{\sqrt{A^2+O^2}}\\&=&\frac{1}{\sqrt{1+\left(\frac{O}{A}\right)^2}}\\&=&\frac{1}{\sqrt{1+\left[\tan\left(\arctan(x)\right)\right]^2}}\\&=&\frac{1}{\sqrt{1+x^2}}\end{eqnarray}$

cos(arccosec(x))
$\begin{eqnarray}\cos\left(\text{arccosec}(x)\right)&=&\frac{A}{H}\\&=&\frac{\sqrt{H^2-O^2}}{H}\\&=&\frac{\sqrt{\left(\frac{H}{O}\right)^2-1}}{\left(\frac{H}{O}\right)}\\&=&\frac{\sqrt{\left[\text{cosec}\left(\text{arccosec}(x)\right)\right]^2-1}}{\text{cosec}\left(\text{arccosec}(x)\right)}\\&=&\frac{\sqrt{x^2-1}}{x}\end{eqnarray}$

cos(arcsec(x))
$\begin{eqnarray}\cos\left(\text{arcsec}(x)\right)&=&\frac{A}{H}\\&=&\frac{1}{\left(\frac{H}{A}\right)}\\&=&\frac{1}{\sec\left(\text{arcsec}(x)\right)}\\&=&\frac{1}{x}\end{eqnarray}$

cos(arccot(x))
$\begin{eqnarray}\cos\left(\text{arccot}(x)\right)&=&\frac{A}{H}\\&=&\frac{A}{\sqrt{O^2+A^2}}\\&=&\frac{\left(\frac{A}{O}\right)}{\sqrt{1+\left(\frac{A}{O}\right)^2}}\\&=&\frac{\text{arccot}\left(\cot(x)\right)}{\sqrt{1+\left[\text{arccot}\left(\cot(x)\right)\right]^2}}\\&=&\frac{x}{\sqrt{1+x^2}}\end{eqnarray}$

tan(arcsin(x))
$\begin{eqnarray}\tan\left(\arcsin(x)\right)&=&\frac{O}{A}\\&=&\frac{O}{\sqrt{H^2-O^2}}\\&=&\frac{\left(\frac{O}{H}\right)}{\sqrt{1-\left(\frac{O}{H}\right)^2}}\\&=&\frac{\sin\left(\arcsin(x)\right)}{\sqrt{1-\left[\sin\left(\arcsin(x)\right)\right]^2}}\\&=&\frac{x}{\sqrt{1-x^2}}\end{eqnarray}$

tan(arccos(x))
$\begin{eqnarray}\tan\left(\arccos(x)\right)&=&\frac{O}{A}\\&=&\frac{\sqrt{H^2-A^2}}{A}\\&=&\frac{\sqrt{1-\left(\frac{A}{H}\right)^2}}{\left(\frac{A}{H}\right)}\\&=&\frac{\sqrt{1-\left[\cos\left(\arccos(x)\right)right]^2}}{\cos\left(\arccos(x)\right)}\\&=&\frac{\sqrt{1-x^2}}{x}\end{eqnarray}$

tan(arccosec(x))
$\begin{eqnarray}\tan\left(\text{arccosec}(x)\right)&=&\frac{O}{A}\\&=&\frac{O}{\sqrt{H^2-O^2}}\\&=&\frac{1}{\sqrt{\left(\frac{H}{O}\right)^2-1}}\\&=&\frac{1}{\sqrt{\left[\text{cosec}\left(\text{arccosec}(x)\right)\right]^2-1}}\\&=&\frac{1}{\sqrt{x^2-1}}\end{eqnarray}$

tan(arcsec(x))
$\begin{eqnarray}\tan\left(\text{arcsec}(x)\right)&=&\frac{O}{A}\\&=&\frac{\sqrt{H^2-A^2}}{A}\\&=&\sqrt{\left(\frac{H}{A}\right)^2-1}\\&=&\sqrt{\left[\sec\left(\text{arcsec}(x)\right)\right]^2-1}\\&=&\sqrt{x^2-1}\end{eqnarray}$

tan(arccot(x))
$\begin{eqnarray}\tan\left(\text{arccot}(x)\right)&=&\frac{O}{A}\\&=&\frac{1}{\left(\frac{A}{O}\right)}\\&=&\frac{1}{\cot\left(\text{arccot}(x)\right)}\\&=&\frac{1}{x}\end{eqnarray}$

cosec(arcsin(x))
$\begin{eqnarray}\text{cosec}\left(\arcsin(x)\right)&=&\frac{H}{O}\\&=&\frac{1}{\left(\frac{O}{H}\right)}\\&=&\frac{1}{\sin\left(\arcsin(x)\right)}\\&=&\frac{1}{x}\end{eqnarray}$

cosec(arccos(x))
$\begin{eqnarray}\text{cosec}\left(\arccos(x)\right)&=&\frac{H}{O}\\&=&\frac{H}{\sqrt{H^2-A^2}}\\&=&\frac{1}{\sqrt{1-\left(\frac{A}{H}\right)^2}}\\&=&\frac{1}{\sqrt{1-\left[\arccos\left(\cos(x)\right)\right]^2}}\\&=&\frac{1}{\sqrt{1-x^2}}\end{eqnarray}$

cosec(arctan(x))
$\begin{eqnarray}\text{cosec}\left(\arctan(x)\right)&=&\frac{H}{O}\\&=&\frac{\sqrt{A^2+O^2}}{O}\\&=&\frac{\sqrt{1+\left(\frac{O}{A}\right)^2}}{\left(\frac{O}{A}\right)}\\&=&\frac{\sqrt{1+\left[\arctan\left(\tan(x)\right)\right]^2}}{\arctan\left(\tan(x)\right)}\\&=&\frac{\sqrt{1+x^2}}{x}\end{eqnarray}$

cosec(arcsec(x))
$\begin{eqnarray}\text{cosec}\left(\text{arcsec}(x)\right)&=&\frac{H}{O}\\&=&\frac{H}{\sqrt{H^2-A^2}}\\&=&\frac{\left(\frac{H}{A}\right)}{\sqrt{\left(\frac{H}{A}\right)^2-1}}\\&=&\frac{\text{arcsec}\left(\sec(x)\right)}{\sqrt{\left[\text{arcsec}\left(\sec(x)\right)\right]^2-1}}\\&=&\frac{x}{\sqrt{x^2-1}}\end{eqnarray}$

cosec(arccot(x))
$\begin{eqnarray}\text{cosec}\left(\text{arccot}(x)\right)&=&\frac{H}{O}\\&=&\frac{\sqrt{O^2+A^2}}{O}\\&=&\sqrt{1+\left(\frac{A}{O}\right)^2}\\&=&\sqrt{1+\left[\cot\left(\text{arccot}(x)\right)\right]^2}\\&=&\sqrt{1+x^2}\end{eqnarray}$

sec(arcsin(x))
$\begin{eqnarray}\sec\left(\arcsin(x)\right)&=&\frac{H}{A}\\&=&\frac{H}{\sqrt{H^2-O^2}}\\&=&\frac{1}{\sqrt{1-\left(\frac{O}{H}\right)^2}}\\&=&\frac{1}{\sqrt{1-\left[\arcsin\left(\sin(x)\right)\right]^2}}\\&=&\frac{1}{\sqrt{1-x^2}}\end{eqnarray}$

sec(arccos(x))
$\begin{eqnarray}\sec\left(\arccos(x)\right)&=&\frac{H}{A}\\&=&\frac{1}{\left(\frac{A}{H}\right)}\\&=&\frac{1}{\arccos\left(\cos(x)\right)}\\&=&\frac{1}{x}\end{eqnarray}$

sec(arctan(x))
$\begin{eqnarray}\sec\left(\arctan(x)\right)&=&\frac{H}{A}\\&=&\frac{\sqrt{A^2+O^2}}{A}\\&=&\sqrt{1+\left(\frac{O}{A}\right)^2}\\&=&\sqrt{1+\left[\arctan\left(\tan(x)\right)\right]^2}\\&=&\sqrt{1+x^2}}\end{eqnarray}$

sec(arccosec(x))
$\begin{eqnarray}\sec\left(\text{arccosec}(x)\right)&=&\frac{H}{A}\\&=&\frac{H}{\sqrt{H^2-O^2}}\\&=&\frac{\left(\frac{H}{O}\right)}{\sqrt{\left(\frac{H}{O}\right)^2-1}}\\&=&\frac{\text{arccosec}\left(\text{cosec}(x)\right)}{\sqrt{\left[\text{arccosec}\left(\text{cosec}(x)\right)\right]^2-1}}\\&=&\frac{x}{\sqrt{x^2-1}}\end{eqnarray}$

sec(arccot(x))
$\begin{eqnarray}\sec\left(\text{arccot}(x)\right)&=&\frac{H}{A}\\&=&\frac{\sqrt{O^2+A^2}}{A}\\&=&\frac{\sqrt{1+\left(\frac{A}{O}\right)^2}}{\left(\frac{A}{O}\right)}\\&=&\frac{\sqrt{1+\left[\text{arccot}\left(\cot(x)\right)\right]^2}}{\text{arccot}\left(\cot(x)\right)}\\&=&\frac{\sqrt{1+x^2}}{x}\end{eqnarray}$

cot(arcsin(x))
$\begin{eqnarray}\cot\left(\arcsin(x)\right)&=&\frac{A}{O}\\&=&\frac{\sqrt{H^2-O^2}}{O}\\&=&\frac{\sqrt{1-\left(\frac{O}{H}\right)^2}}{\left(\frac{O}{H}\right)}\\&=&\frac{\sqrt{1-\left[\arcsin\left(\sin(x)\right)\right]^2}}{\arcsin\left(\sin(x)\right)}\\&=&\frac{\sqrt{1-x^2}}{x}\end{eqnarray}$

cot(arccos(x))
$\begin{eqnarray}\cot\left(\arccos(x)\right)&=&\frac{A}{O}\\&=&\frac{A}{\sqrt{H^2-A^2}}\\&=&\frac{\left(\frac{A}{H}\right)}{\sqrt{1-\left(\frac{A}{H}\right)^2}}\\&=&\frac{\arccos\left(\cos(x)\right)}{\sqrt{1-\left[\arccos\left(\cos(x)\right)\right]^2}}\\&=&\frac{x}{\sqrt{1-x^2}}\end{eqnarray}$

cot(arctan(x))
$\begin{eqnarray}\cot\left(\arctan(x)\right)&=&\frac{A}{O}\\&=&\frac{1}{\left(\frac{O}{A}\right)}\\&=&\frac{1}{\arctan\left(\tan(x)\right)}\\&=&\frac{1}{x}\end{eqnarray}$

cot(arccosec(x))
$\begin{eqnarray}\cot\left(\text{arccosec}(x)\right)&=&\frac{A}{O}\\&=&\frac{\sqrt{H^2-O^2}}{O}\\&=&\sqrt{\left(\frac{H}{O}\right)^2-1}\\&=&\sqrt{\left[\text{arccosec}\left(\text{cosec}(x)\right)\right]^2-1}\\&=&\sqrt{x^2-1}\end{eqnarray}$

cot(arcsec(x))
$\begin{eqnarray}\cot\left(\text{arcsec}(x)\right)&=&\frac{A}{O}\\&=&\frac{A}{\sqrt{H^2-A^2}}\\&=&\frac{1}{\sqrt{\left(\frac{H}{A}\right)^2-1}}\\&=&\frac{1}{\sqrt{\left[\text{arcsec}\left(\sec(x)\right)\right]^2-1}}\\&=&\frac{1}{\sqrt{x^2-1}}\end{eqnarray}$

Hyperbolic functions

Forward-reverse hyperbolic trigonometric formulae
foo(arcbar(x))	arcsinh	arccosh	arctanh	arccosech	arcsech	arccoth
sinh	$x$	$\sqrt{x^2-1}$	$x/\sqrt{1-x^2}$	$1/x$	$\sqrt{1-x^2}/x$	$1/\sqrt{x^2-1}$
cosh	$\sqrt{1+x^2}$	$x$	$1/\sqrt{1-x^2}$	$\sqrt{1+x^2}/x$	$1/x$	$x/\sqrt{x^2-1}$
tanh	$x/\sqrt{1+x^2}$	$\sqrt{x^2-1}/x$	$x$	$1/\sqrt{1+x^2}$	$\sqrt{1-x^2}$	$1/x$
cosech	$1/x$	$1/\sqrt{x^2-1}$	$\sqrt{1-x^2}/x$	$x$	$x/\sqrt{1-x^2}$	$\sqrt{x^2-1}$
sech	$1/\sqrt{1+x^2}$	$1/x$	$\sqrt{1-x^2}$	$x/\sqrt{1+x^2}$	$x$	$\sqrt{x^2-1}/x$
coth	$\sqrt{1+x^2}/x$	$x/\sqrt{x^2-1}$	$1/x$	$\sqrt{1+x^2}$	$1/\sqrt{1-x^2}$	$x$

Proofs for the above formulae

The proofs for the hyperbolic trigonometric formulae use the following identities for the hyperbolic trigonometric functions:

$\sinh(x)=\left(e^x-e^{-x}\right)/2\\\cosh(x)=\left(e^x+e^{-x}\right)/2\\\tanh(x)=\sinh(x)/\cosh(x)=\left(e^x-e^{-x}\right)/\left(e^x+e^{-x}\right)\\\text{cosech}(x)=1/\sinh(x)=2/\left(e^x-e^{-x}\right)\\\text{sech}(x)=1/\cosh(x)=2/\left(e^x+e^{-x}\right)\\\coth(x)=1/\tanh(x)=\left(e^x+e^{-x}\right)/\left(e^x-e^{-x}\right)$

And for the inverse hyperbolic trigonometric functions:

$\text{arcsinh}(x)=\log\left{x+\sqrt{x^2+1}\right}\\\text{arccosh}(x)=\log\left{x+\sqrt{x^2-1}\right}\\\text{arctanh}(x)=\log\left{\sqrt{\frac{1+x}{1-x}}\right}\\\text{arccosech}(x)=\log\left{\frac{1+\sqrt{1+x^2}}{x}\right}\\\text{arcsech}(x)=\log\left{\frac{1+\sqrt{1-x^2}}{x}\right}\\\text{arccoth}(x)=\log\left{\sqrt{\frac{x+1}{x-1}}\right}$

Derivation of the inverse hyperbolic trigonometric functions

arcsinh(x)
$\sinh\left(\text{arcsinh}(x)\right)=x\\e^{\text{arcsinh}(x)}-e^{-\text{arcsinh}(x)}=2x\\e^{2\text{arcsinh}(x)}-2xe^{\text{arcsinh}(x)}-1=0\\\begin{eqnarray}e^{\text{arcsinh}(x)}&=&\frac{2x+\sqrt{4x^2+4}}{2}\\&=&x+\sqrt{x^2+1}\end{eqnarray}\\\text{arcsinh}(x)=\log\left{x+\sqrt{x^2+1}\right}$

arccosh(x)
$\cosh\left(\text{arccosh}(x)\right)=x\\e^{\text{arccosh}(x)}+e^{-\text{arccosh}(x)}=2x\\e^{2\text{arccosh}(x)}-2xe^{\text{arccosh}(x)}+1=0\\\begin{eqnarray}e^{\text{arccosh}(x)}&=&\frac{2x+\sqrt{4x^2-4}}{2}\\&=&x+\sqrt{x^2-1}\end{eqnarray}\\\text{arccosh}(x)=\log\left{x+\sqrt{x^2-1}\right}$

arctanh(x)

$\tanh\left(\text{arctanh}(x)\right)=x\\\frac{e^{\text{arctanh}(x)}-e^{-\text{arctanh}(x)}}{e^{\text{arctanh}(x)}+e^{-\text{arctanh}(x)}}=x\\e^{\text{arctanh}(x)}-e^{-\text{arctanh}(x)}=xe^{\text{arctanh}(x)}+xe^{-\text{arctanh}(x)}\\e^{2\text{arctanh}(x)}-1=xe^{2\text{arctanh}(x)}+x\\(1-x)e^{2\text{arctanh}(x)}=1+x\\e^{2\text{arctanh}(x)}=\frac{1+x}{1-x}\\\text{arctanh}(x)=\log\left{\sqrt{\frac{1+x}{1-x}}\right}$

arccosech(x)
$\text{cosech}\left(\text{arccosech}(x)\right)=x\\\frac{2}{e^{\text{arccosech}(x)}-e^{-\text{arccosech}(x)}}=x\\xe^{\text{arccosech}(x)}-xe^{-\text{arccosech}(x)}=2\\xe^{2\text{arccosech}(x)}-2e^{\text{arccosech}(x)}-x=0\\\begin{eqnarray}e^{\text{arccosech}(x)}&=&\frac{2+\sqrt{4+4x^2}}{2x}\\&=&\frac{1+\sqrt{1+x^2}}{x}\end{eqnarray}\\\text{arccosech}(x)=\log\left{\frac{1+\sqrt{1+x^2}}{x}\right}$

arcsech(x)
$\text{sech}\left(\text{arcsech}(x)\right)=x\\\frac{2}{e^{\text{arcsech}(x)}+e^{-\text{arcsech}(x)}}=x\\xe^{\text{arcsech}(x)}+xe^{-\text{arcsech}(x)}=2\\xe^{2\text{arcsech}(x)}-2e^{\text{arcsech}(x)}+x=0\\\begin{eqnarray}e^{\text{arcsech}(x)}&=&\frac{2+\sqrt{4-4x^2}}{2x}\\&=&\frac{1+\sqrt{1-x^2}}{x}\end{eqnarray}\\\text{arcsech}(x)=\log\left{\frac{1+\sqrt{1-x^2}}{x}\right}$

arccoth(x)

$\coth\left(\text{arccoth}(x)\right)=x\\\frac{e^{\text{arccoth}(x)}+e^{-\text{arccoth}(x)}}{e^{\text{arccoth}(x)}-e^{-\text{arccoth}(x)}}=x\\e^{\text{arccoth}(x)}+e^{-\text{arccoth}(x)}=xe^{\text{arccoth}(x)}-xe^{-\text{arccoth}(x)}\\e^{2\text{arccoth}(x)}+1=xe^{2\text{arccoth}(x)}-x\\(x-1)e^{2\text{arccoth}(x)}=x+1\\e^{2\text{arccoth}(x)}=\frac{x+1}{x-1}\\\text{arccoth}(x)=\log\left{\sqrt{\frac{x+1}{x-1}}\right}$

Together these can be used to derive the forward-reverse hyperbolic trigonometric functions.

sinh(arccosh(x))

$\begin{eqnarray}\sinh\left(\text{arccosh}(x)\right)&=&\frac{e^{\text{arccosh}(x)}-e^{-\text{arccosh}(x)}}{2}\\&=&\frac{e^{\log\left{x+\sqrt{x^2-1}\right}}-e^{-\log\left{x+\sqrt{x^2-1}\right}}}{2}\\&=&\frac{\left(x+\sqrt{x^2-1}\right)-\frac{1}{\left(x+\sqrt{x^2-1}\right)}}{2}\\&=&\frac{x+\sqrt{x^2-1}-\frac{x-\sqrt{x^2-1}}{x^2-\left(x^2-1\right)}}{2}\\&=&\frac{x+\sqrt{x^2-1}-\left(x-\sqrt{x^2-1}\right)}{2}\\&=&\sqrt{x^2-1}\end{eqnarray}$

sinh(arctanh(x))

$\begin{eqnarray}\sinh\left(\text{arctanh}(x)\right)&=&\frac{e^{\text{arctanh}(x)}-e^{-\text{arctanh}(x)}}{2}\\&=&\frac{e^{\log\left{\sqrt{\frac{1+x}{1-x}}\right}}-e^{-\log\left{\sqrt{\frac{1+x}{1-x}}\right}}}{2}\\&=&\frac{\left(\sqrt{\frac{1+x}{1-x}}\right)-\left(\sqrt{\frac{1-x}{1+x}}\right)}{2}\\&=&\frac{\sqrt{\frac{(1+x)^2}{1-x^2}}-\sqrt{\frac{(1-x)^2}{1-x^2}}}{2}\\&=&\frac{\frac{(1+x)-(1-x)}{\sqrt{1-x^2}}}{2}\\&=&\frac{x}{\sqrt{1-x^2}}\end{eqnarray}$

sinh(arccosech(x))

$\begin{eqnarray}\sinh\left(\text{arccosech}(x)\right)&=&\frac{e^{\text{arccosech}(x)}-e^{-\text{arccosech}(x)}}{2}\\&=&\frac{e^{\log\left{\frac{1+\sqrt{1+x^2}}{x}\right}}-e^{-\log\left{\frac{1+\sqrt{1+x^2}}{x}\right}}}{2}\\&=&\frac{\left(\frac{1+\sqrt{1+x^2}}{x}\right)-\left(\frac{x}{1+\sqrt{1+x^2}}\right)}{2}\\&=&\frac{\frac{1+\sqrt{1+x^2}}{x}-\frac{x\left(1-\sqrt{1+x^2}\right)}{1-\left(1+x^2\right)}}{2}\\&=&\frac{\frac{1+\sqrt{1+x^2}}{x}+\frac{1-\sqrt{1+x^2}}{x}}{2}\\&=&\frac{1}{x}\end{eqnarray}$

sinh(arcsech(x))

$\begin{eqnarray}\sinh\left(\text{arcsech}(x)\right)&=&\frac{e^{\text{arcsech}(x)}-e^{-\text{arcsech}(x)}}{2}\\&=&\frac{e^{\log\left{\frac{1+\sqrt{1-x^2}}{x}\right}}-e^{-\log\left{\frac{1+\sqrt{1-x^2}}{x}\right}}}{2}\\&=&\frac{\left(\frac{1+\sqrt{1-x^2}}{x}\right)-\left(\frac{x}{1+\sqrt{1-x^2}}\right)}{2}\\&=&\frac{\frac{1+\sqrt{1-x^2}}{x}-\frac{x\left(1-\sqrt{1-x^2}\right)}{1-\left(1-x^2\right)}}{2}\\&=&\frac{\frac{1+\sqrt{1-x^2}}{x}-\frac{1-\sqrt{1-x^2}}{x}}{2}\\&=&\frac{\sqrt{1-x^2}}{x}\end{eqnarray}$

sinh(arccoth(x))

$\begin{eqnarray}\sinh\left(\text{arccoth}(x)\right)&=&\frac{e^{\text{arccoth}(x)}-e^{-\text{arccoth}(x)}}{2}\\&=&\frac{e^{\log\left{\sqrt{\frac{x+1}{x-1}}\right}}-e^{-\log\left{\sqrt{\frac{x+1}{x-1}}\right}}}{2}\\&=&\frac{\left(\sqrt{\frac{x+1}{x-1}}\right)-\left(\sqrt{\frac{x-1}{x+1}}\right)}{2}\\&=&\frac{\sqrt{\frac{(x+1)^2}{x^2-1}}-\sqrt{\frac{(x-1)^2}{x^2-1}}}{2}\\&=&\frac{\frac{x+1}{\sqrt{x^2-1}}-\frac{x-1}{\sqrt{x^2-1}}}{2}\\&=&\frac{1}{\sqrt{x^2-1}}\end{eqnarray}$

cosh(arcsinh(x))

$\begin{eqnarray}\cosh\left(\text{arcsinh}(x)\right)&=&\frac{e^{\text{arcsinh}(x)}+e^{-\text{arcsinh}(x)}}{2}\\&=&\frac{e^{\log\left{x+\sqrt{x^2+1}\right}}+e^{-\log\left{x+\sqrt{x^2+1}\right}}}{2}\\&=&\frac{\left(x+\sqrt{x^2+1}\right)+\frac{1}{\left(x+\sqrt{x^2+1}\right)}}{2}\\&=&\frac{x+\sqrt{x^2+1}+\frac{x-\sqrt{x^2+1}}{x^2-\left(x^2+1\right)}}{2}\\&=&\frac{x+\sqrt{x^2+1}-\left(x-\sqrt{x^2+1}\right)}{2}\\&=&\sqrt{x^2+1}\end{eqnarray}$

cosh(arctanh(x))

$\begin{eqnarray}\cosh\left(\text{arctanh}(x)\right)&=&\frac{e^{\text{arctanh}(x)}+e^{-\text{arctanh}(x)}}{2}\\&=&\frac{e^{\log\left{\sqrt{\frac{1+x}{1-x}}\right}}+e^{-\log\left{\sqrt{\frac{1+x}{1-x}}\right}}}{2}\\&=&\frac{\left(\sqrt{\frac{1+x}{1-x}}\right)+\left(\sqrt{\frac{1-x}{1+x}}\right)}{2}\\&=&\frac{\sqrt{\frac{(1+x)^2}{1-x^2}}+\sqrt{\frac{(1-x)^2}{1-x^2}}}{2}\\&=&\frac{\frac{1+x}{\sqrt{1-x^2}}+\frac{1-x}{\sqrt{1-x^2}}}{2}\\&=&\frac{1}{\sqrt{1-x^2}}\end{eqnarray}$

cosh(arccosech(x))

$\begin{eqnarray}\cosh\left(\text{arccosech}(x)\right)&=&\frac{e^{\text{arccosech}(x)}+e^{-\text{arccosech}(x)}}{2}\\&=&\frac{e^{\log\left{\frac{1+\sqrt{1+x^2}}{x}\right}}+e^{-\log\left{\frac{1+\sqrt{1+x^2}}{x}\right}}}{2}\\&=&\frac{\left(\frac{1+\sqrt{1+x^2}}{x}\right)+\left(\frac{x}{1+\sqrt{1+x^2}}\right)}{2}\\&=&\frac{\frac{1+\sqrt{1+x^2}}{x}+\frac{x\left(1-\sqrt{1+x^2}\right)}{1-\left(1+x^2\right)}}{2}\\&=&\frac{\frac{1+\sqrt{1+x^2}}{x}-\frac{1-\sqrt{1+x^2}}{x}}{2}\\&=&\frac{\sqrt{1+x^2}}{x}\end{eqnarray}$

cosh(arcsech(x))

$\begin{eqnarray}\cosh\left(\text{arcsech}(x)\right)&=&\frac{e^{\text{arcsech}(x)}+e^{-\text{arcsech}(x)}}{2}\\&=&\frac{e^{\log\left{\frac{1+\sqrt{1-x^2}}{x}\right}}+e^{-\log\left{\frac{1+\sqrt{1-x^2}}{x}\right}}}{2}\\&=&\frac{\left(\frac{1+\sqrt{1-x^2}}{x}\right)+\left(\frac{x}{1+\sqrt{1-x^2}}\right)}{2}\\&=&\frac{\frac{1+\sqrt{1-x^2}}{x}+\frac{x\left(1-\sqrt{1-x^2}\right)}{1-\left(1-x^2\right)}}{2}\\&=&\frac{\frac{1+\sqrt{1-x^2}}{x}+\frac{1-\sqrt{1-x^2}}{x}}{2}\\&=&\frac{1}{x}\end{eqnarray}$

cosh(arccoth(x))

$\begin{eqnarray}\cosh\left(\text{arccoth}(x)\right)&=&\frac{e^{\text{arccoth}(x)}+e^{-\text{arccoth}(x)}}{2}\\&=&\frac{e^{\log\left{\sqrt{\frac{x+1}{x-1}}\right}}+e^{-\log\left{\sqrt{\frac{x+1}{x-1}}\right}}}{2}\\&=&\frac{\left(\sqrt{\frac{x+1}{x-1}}\right)+\left(\sqrt{\frac{x-1}{x+1}}\right)}{2}\\&=&\frac{\sqrt{\frac{(x+1)^2}{x^2-1}}+\sqrt{\frac{(x-1)^2}{x^2-1}}}{2}\\&=&\frac{\frac{x+1}{\sqrt{x^2-1}}+\frac{x-1}{\sqrt{x^2-1}}}{2}\\&=&\frac{x}{\sqrt{x^2-1}}\end{eqnarray}$

tanh(arcsinh(x))

$\begin{eqnarray}\tanh\left(\text{arcsinh}(x)\right)&=&\frac{e^{\text{arcsinh}(x)}-e^{-\text{arcsinh}(x)}}{e^{\text{arcsinh}(x)}+e^{-\text{arcsinh}(x)}}\\&=&\frac{e^{\log\left{x+\sqrt{x^2+1}\right}}-e^{-\log\left{x+\sqrt{x^2+1}\right}}}{e^{\log\left{x+\sqrt{x^2+1}\right}}+e^{-\log\left{x+\sqrt{x^2+1}\right}}}\\&=&\frac{\left(x+\sqrt{x^2+1}\right)-\frac{1}{\left(x+\sqrt{x^2+1}\right)}}{\left(x+\sqrt{x^2+1}\right)+\frac{1}{\left(x+\sqrt{x^2+1}\right)}}\\&=&\frac{x+\sqrt{x^2+1}-\frac{x-\sqrt{x^2+1}}{x^2-\left(x^2+1\right)}}{x+\sqrt{x^2+1}+\frac{x-\sqrt{x^2+1}}{x^2-\left(x^2+1\right)}}\\&=&\frac{x+\sqrt{x^2+1}+\left(x-\sqrt{x^2+1}\right)}{x+\sqrt{x^2+1}-\left(x-\sqrt{x^2+1}\right)}\\&=&\frac{x}{\sqrt{x^2+1}}\end{eqnarray}$

tanh(arccosh(x))

$\begin{eqnarray}\tanh\left(\text{arccosh}(x)\right)&=&\frac{e^{\text{arccosh}(x)}-e^{-\text{arccosh}(x)}}{e^{\text{arccosh}(x)}+e^{-\text{arccosh}(x)}}\\&=&\frac{e^{\log\left{x+\sqrt{x^2-1}\right}}-e^{-\log\left{x+\sqrt{x^2-1}\right}}}{e^{\log\left{x+\sqrt{x^2-1}\right}}+e^{-\log\left{x+\sqrt{x^2-1}\right}}}\\&=&\frac{\left(x+\sqrt{x^2-1}\right)-\frac{1}{\left(x+\sqrt{x^2-1}\right)}}{\left(x+\sqrt{x^2-1}\right)+\frac{1}{\left(x+\sqrt{x^2-1}\right)}}\\&=&\frac{x+\sqrt{x^2-1}-\frac{x-\sqrt{x^2-1}}{x^2-\left(x^2-1\right)}}{x+\sqrt{x^2-1}+\frac{x-\sqrt{x^2-1}}{x^2-\left(x^2-1\right)}}\\&=&\frac{x+\sqrt{x^2-1}-\left(x-\sqrt{x^2-1}\right)}{x+\sqrt{x^2-1}+\left(x-\sqrt{x^2-1}\right)}\\&=&\frac{\sqrt{x^2-1}}{x}\end{eqnarray}$

tanh(arccosech(x))

$\begin{eqnarray}\tanh\left(\text{arccosech}(x)\right)&=&\frac{e^{\text{arccosech}(x)}-e^{-\text{arccosech}(x)}}{e^{\text{arccosech}(x)}+e^{-\text{arccosech}(x)}}\\&=&\frac{e^{\log\left{\frac{1+\sqrt{1+x^2}}{x}\right}}-e^{-\log\left{\frac{1+\sqrt{1+x^2}}{x}\right}}}{e^{\log\left{\frac{1+\sqrt{1+x^2}}{x}\right}}+e^{-\log\left{\frac{1+\sqrt{1+x^2}}{x}\right}}}\\&=&\frac{\left(\frac{1+\sqrt{1+x^2}}{x}\right)-\left(\frac{x}{1+\sqrt{1+x^2}}\right)}{\left(\frac{1+\sqrt{1+x^2}}{x}\right)+\left(\frac{x}{1+\sqrt{1+x^2}}\right)}\\&=&\frac{\frac{1+\sqrt{1+x^2}}{x}-\frac{x\left(1-\sqrt{1+x^2}\right)}{1-\left(1+x^2\right)}}{\frac{1+\sqrt{1+x^2}}{x}+\frac{x\left(1-\sqrt{1+x^2}\right)}{1-\left(1+x^2\right)}}\\&=&\frac{\frac{1+\sqrt{1+x^2}}{x}+\frac{1-\sqrt{1+x^2}}{x}}{\frac{1+\sqrt{1+x^2}}{x}-\frac{1-\sqrt{1+x^2}}{x}}\\&=&\frac{1}{\sqrt{1+x^2}}\end{eqnarray}$

tanh(arcsech(x))

$\begin{eqnarray}\tanh\left(\text{arcsech}(x)\right)&=&\frac{e^{\text{arcsech}(x)}-e^{-\text{arcsech}(x)}}{e^{\text{arcsech}(x)}+e^{-\text{arcsech}(x)}}\\&=&\frac{e^{\log\left{\frac{1+\sqrt{1-x^2}}{x}\right}}-e^{-\log\left{\frac{1+\sqrt{1-x^2}}{x}\right}}}{e^{\log\left{\frac{1+\sqrt{1-x^2}}{x}\right}}+e^{-\log\left{\frac{1+\sqrt{1-x^2}}{x}\right}}}\\&=&\frac{\left(\frac{1+\sqrt{1-x^2}}{x}\right)-\left(\frac{x}{1+\sqrt{1-x^2}}\right)}{\left(\frac{1+\sqrt{1-x^2}}{x}\right)+\left(\frac{x}{1+\sqrt{1-x^2}}\right)}\\&=&\frac{\frac{1+\sqrt{1-x^2}}{x}-\frac{x\left(1-\sqrt{1-x^2}\right)}{1-\left(1-x^2\right)}}{\frac{1+\sqrt{1-x^2}}{x}+\frac{x\left(1-\sqrt{1-x^2}\right)}{1-\left(1-x^2\right)}}\\&=&\frac{\frac{1+\sqrt{1-x^2}}{x}-\frac{1-\sqrt{1-x^2}}{x}}{\frac{1+\sqrt{1-x^2}}{x}+\frac{1-\sqrt{1-x^2}}{x}}\\&=&\sqrt{1-x^2}\end{eqnarray}$

tanh(arccoth(x))

$\begin{eqnarray}\tanh\left(\text{arccoth}(x)\right)&=&\frac{e^{\text{arccoth}(x)}-e^{-\text{arccoth}(x)}}{e^{\text{arccoth}(x)}+e^{-\text{arccoth}(x)}}\\&=&\frac{e^{\log\left{\sqrt{\frac{x+1}{x-1}}\right}}-e^{-\log\left{\sqrt{\frac{x+1}{x-1}}\right}}}{e^{\log\left{\sqrt{\frac{x+1}{x-1}}\right}}+e^{-\log\left{\sqrt{\frac{x+1}{x-1}}\right}}}\\&=&\frac{\left(\sqrt{\frac{x+1}{x-1}}\right)-\left(\sqrt{\frac{x-1}{x+1}}\right)}{\left(\sqrt{\frac{x+1}{x-1}}\right)+\left(\sqrt{\frac{x-1}{x+1}}\right)}\\&=&\frac{\sqrt{\frac{(x+1)^2}{x^2-1}}-\sqrt{\frac{(x-1)^2}{x^2-1}}}{\sqrt{\frac{(x+1)^2}{x^2-1}}+\sqrt{\frac{(x-1)^2}{x^2-1}}}\\&=&\frac{\frac{x+1}{\sqrt{x^2-1}}-\frac{x-1}{\sqrt{x^2-1}}}{\frac{x+1}{\sqrt{x^2-1}}+\frac{x-1}{\sqrt{x^2-1}}}\\&=&\frac{1}{x}\end{eqnarray}$

cosech(arcsinh(x))

$\begin{eqnarray}\text{cosech}\left(\text{arcsinh}(x)\right)&=&\frac{2}{e^{\text{arcsinh}(x)}-e^{-\text{arcsinh}(x)}}\\&=&\frac{2}{e^{\log\left{x+\sqrt{x^2+1}\right}}-e^{-\log\left{x+\sqrt{x^2+1}\right}}}\\&=&\frac{2}{\left(x+\sqrt{x^2+1}\right)-\frac{1}{\left(x+\sqrt{x^2+1}\right)}}\\&=&\frac{2}{x+\sqrt{x^2+1}-\frac{x-\sqrt{x^2+1}}{x^2-\left(x^2+1\right)}}\\&=&\frac{2}{x+\sqrt{x^2+1}+\left(x-\sqrt{x^2+1}\right)}\\&=&\frac{1}{x}\end{eqnarray}$

cosech(arccosh(x))

$\begin{eqnarray}\text{cosech}\left(\text{arccosh}(x)\right)&=&\frac{2}{e^{\text{arccosh}(x)}-e^{-\text{arccosh}(x)}}\\&=&\frac{2}{e^{\log\left{x+\sqrt{x^2-1}\right}}-e^{-\log\left{x+\sqrt{x^2-1}\right}}}\\&=&\frac{2}{\left(x+\sqrt{x^2-1}\right)-\frac{1}{\left(x+\sqrt{x^2-1}\right)}}\\&=&\frac{2}{x+\sqrt{x^2-1}-\frac{x-\sqrt{x^2-1}}{x^2-\left(x^2-1\right)}}\\&=&\frac{2}{x+\sqrt{x^2-1}-\left(x-\sqrt{x^2-1}\right)}\\&=&\frac{1}{\sqrt{x^2-1}}\end{eqnarray}$

cosech(arctanh(x))

$\begin{eqnarray}\text{cosech}\left(\text{arctanh}(x)\right)&=&\frac{2}{e^{\text{arctanh}(x)}-e^{-\text{arctanh}(x)}}\\&=&\frac{2}{e^{\log\left{\sqrt{\frac{1+x}{1-x}}\right}}-e^{-\log\left{\sqrt{\frac{1+x}{1-x}}\right}}}\\&=&\frac{2}{\left(\sqrt{\frac{1+x}{1-x}}\right)-\left(\sqrt{\frac{1-x}{1+x}}\right)}\\&=&\frac{2}{\sqrt{\frac{(1+x)^2}{1-x^2}}-\sqrt{\frac{(1-x)^2}{1-x^2}}}\\&=&\frac{2}{\frac{1+x}{\sqrt{1-x^2}}-\frac{1-x}{\sqrt{1-x^2}}}\\&=&\frac{\sqrt{1-x^2}}{x}\end{eqnarray}$

cosech(arcsech(x))

$\begin{eqnarray}\text{cosech}\left(\text{arcsech}(x)\right)&=&\frac{2}{e^{\text{arcsech}(x)}-e^{-\text{arcsech}(x)}}\\&=&\frac{2}{e^{\log\left{\frac{1+\sqrt{1-x^2}}{x}\right}}-e^{-\log\left{\frac{1+\sqrt{1-x^2}}{x}\right}}}\\&=&\frac{2}{\left(\frac{1+\sqrt{1-x^2}}{x}\right)-\left(\frac{x}{1+\sqrt{1-x^2}}\right)}\\&=&\frac{2}{\frac{1+\sqrt{1-x^2}}{x}-\frac{x\left(1-\sqrt{1-x^2}\right)}{1-\left(1-x^2\right)}}\\&=&\frac{2}{\frac{1+\sqrt{1-x^2}}{x}-\frac{1-\sqrt{1-x^2}}{x}}\\&=&\frac{x}{\sqrt{1-x^2}}\end{eqnarray}$

cosech(arccoth(x))

$\begin{eqnarray}\text{cosech}\left(\text{arccoth}(x)\right)&=&\frac{2}{e^{\text{arccoth}(x)}-e^{-\text{arccoth}(x)}}\\&=&\frac{2}{e^{\log\left{\sqrt{\frac{x+1}{x-1}}\right}}-e^{-\log\left{\sqrt{\frac{x+1}{x-1}}\right}}}\\&=&\frac{2}{\left(\sqrt{\frac{x+1}{x-1}}\right)-\left(\sqrt{\frac{x-1}{x+1}}\right)}\\&=&\frac{2}{\sqrt{\frac{(x+1)^2}{x^2-1}}-\sqrt{\frac{(x-1)^2}{x^2-1}}}\\&=&\frac{2}{\frac{x+1}{\sqrt{x^2-1}}-\frac{x-1}{\sqrt{x^2-1}}}\\&=&\sqrt{x^2-1}\end{eqnarray}$

sech(arcsinh(x))

$\begin{eqnarray}\text{sech}\left(\text{arcsinh}(x)\right)&=&\frac{2}{e^{\text{arcsinh}(x)}+e^{-\text{arcsinh}(x)}}\\&=&\frac{2}{e^{\log\left{x+\sqrt{x^2+1}\right}}+e^{-\log\left{x+\sqrt{x^2+1}\right}}}\\&=&\frac{2}{\left(x+\sqrt{x^2+1}\right)+\frac{1}{\left(x+\sqrt{x^2+1}\right)}}\\&=&\frac{2}{x+\sqrt{x^2+1}+\frac{x-\sqrt{x^2+1}}{x^2-\left(x^2+1\right)}}\\&=&\frac{2}{x+\sqrt{x^2+1}-\left(x-\sqrt{x^2+1}\right)}\\&=&\frac{1}{\sqrt{x^2+1}}\end{eqnarray}$

sech(arccosh(x))

$\begin{eqnarray}\text{sech}\left(\text{arccosh}(x)\right)&=&\frac{2}{e^{\text{arccosh}(x)}+e^{-\text{arccosh}(x)}}\\&=&\frac{2}{e^{\log\left{x+\sqrt{x^2-1}\right}}+e^{-\log\left{x+\sqrt{x^2-1}\right}}}\\&=&\frac{2}{\left(x+\sqrt{x^2-1}\right)+\frac{1}{\left(x+\sqrt{x^2-1}\right)}}\\&=&\frac{2}{x+\sqrt{x^2-1}+\frac{x-\sqrt{x^2-1}}{x^2-\left(x^2-1\right)}}\\&=&\frac{2}{x+\sqrt{x^2-1}+\left(x-\sqrt{x^2-1}\right)}\\&=&\frac{1}{x}\end{eqnarray}$

sech(arctanh(x))

$\begin{eqnarray}\text{sech}\left(\text{arctanh}(x)\right)&=&\frac{2}{e^{\text{arctanh}(x)}+e^{-\text{arctanh}(x)}}\\&=&\frac{2}{e^{\log\left{\sqrt{\frac{1+x}{1-x}}\right}}+e^{-\log\left{\sqrt{\frac{1+x}{1-x}}\right}}}\\&=&\frac{2}{\left(\sqrt{\frac{1+x}{1-x}}\right)+\left(\sqrt{\frac{1-x}{1+x}}\right)}\\&=&\frac{2}{\sqrt{\frac{(1+x)^2}{1-x^2}}+\sqrt{\frac{(1-x)^2}{1-x^2}}}\\&=&\frac{2}{\frac{1+x}{\sqrt{1-x^2}}+\frac{1-x}{\sqrt{1-x^2}}}\\&=&\sqrt{1-x^2}\end{eqnarray}$

sech(arccosech(x))

$\begin{eqnarray}\text{sech}\left(\text{arccosech}(x)\right)&=&\frac{2}{e^{\text{arccosech}(x)}+e^{-\text{arccosech}(x)}}\\&=&\frac{2}{e^{\log\left{\frac{1+\sqrt{1+x^2}}{x}\right}}+e^{-\log\left{\frac{1+\sqrt{1+x^2}}{x}\right}}}\\&=&\frac{2}{\left(\frac{1+\sqrt{1+x^2}}{x}\right)+\left(\frac{x}{1+\sqrt{1+x^2}}\right)}\\&=&\frac{2}{\frac{1+\sqrt{1+x^2}}{x}+\frac{x\left(1-\sqrt{1+x^2}\right)}{1-\left(1+x^2\right)}}\\&=&\frac{2}{\frac{1+\sqrt{1+x^2}}{x}-\frac{1-\sqrt{1+x^2}}{x}}\\&=&\frac{x}{\sqrt{1+x^2}}\end{eqnarray}$

sech(arccoth(x))

$\begin{eqnarray}\text{sech}\left(\text{arccoth}(x)\right)&=&\frac{2}{e^{\text{arccoth}(x)}+e^{-\text{arccoth}(x)}}\\&=&\frac{2}{e^{\log\left{\sqrt{\frac{x+1}{x-1}}\right}}+e^{-\log\left{\sqrt{\frac{x+1}{x-1}}\right}}}\\&=&\frac{2}{\left(\sqrt{\frac{x+1}{x-1}}\right)+\left(\sqrt{\frac{x-1}{x+1}}\right)}\\&=&\frac{2}{\sqrt{\frac{(x+1)^2}{x^2-1}}+\sqrt{\frac{(x-1)^2}{x^2-1}}}\\&=&\frac{2}{\frac{x+1}{\sqrt{x^2-1}}+\frac{x-1}{\sqrt{x^2-1}}}\\&=&\frac{\sqrt{x^2-1}}{x}\end{eqnarray}$

coth(arcsinh(x))

$\begin{eqnarray}\coth\left(\text{arcsinh}(x)\right)&=&\frac{e^{\text{arcsinh}(x)}+e^{-\text{arcsinh}(x)}}{e^{\text{arcsinh}(x)}-e^{-\text{arcsinh}(x)}}\\&=&\frac{e^{\log\left{x+\sqrt{x^2+1}\right}}+e^{-\log\left{x+\sqrt{x^2+1}\right}}}{e^{\log\left{x+\sqrt{x^2+1}\right}}-e^{-\log\left{x+\sqrt{x^2+1}\right}}}\\&=&\frac{\left(x+\sqrt{x^2+1}\right)+\frac{1}{\left(x+\sqrt{x^2+1}\right)}}{\left(x+\sqrt{x^2+1}\right)-\frac{1}{\left(x+\sqrt{x^2+1}\right)}}\\&=&\frac{x+\sqrt{x^2+1}+\frac{x-\sqrt{x^2+1}}{x^2-\left(x^2+1\right)}}{x+\sqrt{x^2+1}-\frac{x-\sqrt{x^2+1}}{x^2-\left(x^2+1\right)}}\\&=&\frac{x+\sqrt{x^2+1}-\left(x-\sqrt{x^2+1}\right)}{x+\sqrt{x^2+1}+\left(x-\sqrt{x^2+1}\right)}\\&=&\frac{\sqrt{x^2+1}}{x}\end{eqnarray}$

coth(arccosh(x))

$\begin{eqnarray}\coth\left(\text{arccosh}(x)\right)&=&\frac{e^{\text{arccosh}(x)}+e^{-\text{arccosh}(x)}}{e^{\text{arccosh}(x)}-e^{-\text{arccosh}(x)}}\\&=&\frac{e^{\log\left{x+\sqrt{x^2-1}\right}}+e^{-\log\left{x+\sqrt{x^2-1}\right}}}{e^{\log\left{x+\sqrt{x^2-1}\right}}-e^{-\log\left{x+\sqrt{x^2-1}\right}}}\\&=&\frac{\left(x+\sqrt{x^2-1}\right)+\frac{1}{\left(x+\sqrt{x^2-1}\right)}}{\left(x+\sqrt{x^2-1}\right)-\frac{1}{\left(x+\sqrt{x^2-1}\right)}}\\&=&\frac{x+\sqrt{x^2-1}+\frac{x-\sqrt{x^2-1}}{x^2-\left(x^2-1\right)}}{x+\sqrt{x^2-1}-\frac{x-\sqrt{x^2-1}}{x^2-\left(x^2-1\right)}}\\&=&\frac{x+\sqrt{x^2-1}+\left(x-\sqrt{x^2-1}\right)}{x+\sqrt{x^2-1}-\left(x-\sqrt{x^2-1}\right)}\\&=&\frac{x}{\sqrt{x^2-1}}\end{eqnarray}$

coth(arctanh(x))

$\begin{eqnarray}\coth\left(\text{arctanh}(x)\right)&=&\frac{e^{\text{arctanh}(x)}+e^{-\text{arctanh}(x)}}{e^{\text{arctanh}(x)}-e^{-\text{arctanh}(x)}}\\&=&\frac{e^{\log\left{\sqrt{\frac{1+x}{1-x}}\right}}+e^{-\log\left{\sqrt{\frac{1+x}{1-x}}\right}}}{e^{\log\left{\sqrt{\frac{1+x}{1-x}}\right}}-e^{-\log\left{\sqrt{\frac{1+x}{1-x}}\right}}}\\&=&\frac{\left(\sqrt{\frac{1+x}{1-x}}\right)+\left(\sqrt{\frac{1-x}{1+x}}\right)}{\left(\sqrt{\frac{1+x}{1-x}}\right)-\left(\sqrt{\frac{1-x}{1+x}}\right)}\\&=&\frac{\sqrt{\frac{(1+x)^2}{1-x^2}}+\sqrt{\frac{(1-x)^2}{1-x^2}}}{\sqrt{\frac{(1+x)^2}{1-x^2}}-\sqrt{\frac{(1-x)^2}{1-x^2}}}\\&=&\frac{\frac{1+x}{\sqrt{1-x^2}}+\frac{1-x}{\sqrt{1-x^2}}}{\frac{1+x}{\sqrt{1-x^2}}-\frac{1-x}{\sqrt{1-x^2}}}\\&=&\frac{1}{x}\end{eqnarray}$

coth(arccosech(x))

$\begin{eqnarray}\coth\left(\text{arccosech}(x)\right)&=&\frac{e^{\text{arccosech}(x)}+e^{-\text{arccosech}(x)}}{e^{\text{arccosech}(x)}-e^{-\text{arccosech}(x)}}\\&=&\frac{e^{\log\left{\frac{1+\sqrt{1+x^2}}{x}\right}}+e^{-\log\left{\frac{1+\sqrt{1+x^2}}{x}\right}}}{e^{\log\left{\frac{1+\sqrt{1+x^2}}{x}\right}}-e^{-\log\left{\frac{1+\sqrt{1+x^2}}{x}\right}}}\\&=&\frac{\left(\frac{1+\sqrt{1+x^2}}{x}\right)+\left(\frac{x}{1+\sqrt{1+x^2}}\right)}{\left(\frac{1+\sqrt{1+x^2}}{x}\right)-\left(\frac{x}{1+\sqrt{1+x^2}}\right)}\\&=&\frac{\frac{1+\sqrt{1+x^2}}{x}+\frac{x\left(\1-\sqrt{1+x^2}\right)}{1-\left(1+x^2\right)}}{\frac{1+\sqrt{1+x^2}}{x}-\frac{x\left(1-\sqrt{1+x^2}\right)}{1-\left(1+x^2\right)}}\\&=&\frac{\frac{1+\sqrt{1+x^2}}{x}-\frac{1-\sqrt{1+x^2}}{x}}{\frac{1+\sqrt{1+x^2}}{x}+\frac{1-\sqrt{1+x^2}}{x}}\\&=&\sqrt{1+x^2}\end{eqnarray}$

coth(arcsech(x))

$\begin{eqnarray}\coth\left(\text{arcsech}(x)\right)&=&\frac{e^{\text{arcsech}(x)}+e^{-\text{arcsech}(x)}}{e^{\text{arcsech}(x)}-e^{-\text{arcsech}(x)}}\\&=&\frac{e^{\log\left{\frac{1+\sqrt{1-x^2}}{x}\right}}+e^{-\log\left{\frac{1+\sqrt{1-x^2}}{x}\right}}}{e^{\log\left{\frac{1+\sqrt{1-x^2}}{x}\right}}-e^{-\log\left{\frac{1+\sqrt{1-x^2}}{x}\right}}}\\&=&\frac{\left(\frac{1+\sqrt{1-x^2}}{x}\right)+\left(\frac{x}{1+\sqrt{1-x^2}}\right)}{\left(\frac{1+\sqrt{1-x^2}}{x}\right)-\left(\frac{x}{1+\sqrt{1-x^2}}\right)}\\&=&\frac{\frac{1+\sqrt{1-x^2}}{x}+\frac{x\left(1-\sqrt{1-x^2}\right)}{1-\left(1-x^2\right)}}{\frac{1+\sqrt{1-x^2}}{x}-\frac{x\left(1-\sqrt{1-x^2}\right)}{1-\left(1-x^2\right)}}\\&=&\frac{\frac{1+\sqrt{1-x^2}}{x}+\frac{1-\sqrt{1-x^2}}{x}}{\frac{1+\sqrt{1-x^2}}{x}-\frac{1-\sqrt{1-x^2}}{x}}\\&=&\frac{1}{\sqrt{1-x^2}}\end{eqnarray}$

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