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With complex arguments we come across identities that have a mix, as in $\sin(x+iy)=\sin x \cosh y + i \cos x \sinh y$. But are there any identities that have only real arguments and a mix of functions?

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There are some identities induced by the Gudermannian function, see http://dlmf.nist.gov/4.23#viii

$$\mathrm{gd}\left(x\right) = 2\mathrm{arctan}\left(e^{x}\right) - \tfrac{1}{2}\pi =\mathrm{arcsin}\left(\tanh x\right) =\mathrm{arccsc}\left(\coth x \right)\\ =\mathrm{arccos}\left(\mathrm{sech}x\right) =\mathrm{arcsec} \left(\cosh x\right) =\mathrm{arctan}\left(\sinh x\right) =\mathrm{arccot}\left(\mathrm{csch}x\right)$$

and its inverse

$${\mathrm{gd}^{-1}}\left(x\right) =\ln\tan\left(\tfrac{1}{2}x+\tfrac{1}{4}\pi\right) =\ln\left(\sec x+\tan x\right)=\mathrm{arcsinh}\left(\tan x \right) =\mathrm{arccsch}\left(\cot x\right)\\ =\mathrm{arccosh}\left(\sec x\right) =\mathrm{arcsech}\left(\cos x\right) =\mathrm{arctanh}\left(\sin x\right) =\mathrm{arccoth}\left(\csc x\right).$$

See also https://en.wikipedia.org/wiki/Gudermannian_function#Properties

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