Hyperbolic Fourier type infinite series for $\log(\sinh x)$ and $\log(\cosh x)$ analogous to Fourier $\cos$ series for $\log(\sin x)$ and $\log(\cos x)$
$$\log(\sinh(x))=-\frac{1}{2} (i \pi )-\log (2)-\sum _{k=1}^{\infty } \frac{\cosh (2 k x)}{k} \tag{1}$$
$$\log(\cosh(x))=-\log (2)-\sum _{k=1}^{\infty } \frac{(-1)^k \cosh (2 k x)}{k} \tag{2}$$
These analogous formula were found using Mathematica, but I am not sure about the proof. Do I convert $\cosh(2kx)$ into its exponential form and proceed that way?
Hint:
$$\sin(ix)=i\sinh(x)$$ and $$\cos(ix)=\cosh(x)$$