Are there real identities involving both circular and hyperbolic trig functions? 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?

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
