How to prove that $$ f(x,y)=\cos(\cos(x)\cosh(y))\cosh(\sin(x)\sinh(y)) $$ is harmonic?
What properties of harmonic functions are useful here?
I can express \begin{align} f(x,y)&=1/2 (\cos(\cos(x) \cosh(y) - i \sin(x) \sinh(y)) + \cos(\cos(x) \cosh(y) + i \sin(x) \sinh(y)))\\ &=1/2(\cos(\cos(x+iy))+\cos(\cos(x-iy))\;, \end{align} is there a way to proceed from here?
Use that $\cosh{t}=\cos{it}$, and $\sinh{t}=i\sin{-it}$ so that $2f(x,y)=\cos(\cos{x}\cos{iy}-i\sin{x}i\sin{iy})+\cos(\cos{x}\cos{iy}+i\sin{x}i\sin{iy})=\cos{\cos(x-iy)}+\cos{\cos(x+iy)}$ is the sum of a holomorphic and a anti-holomorphic function, so is harmonic.