Proving $ 2\arcsin\left(\sinh(x)\right) + \arccos\left( 2 - \cosh(2x) \right) = 0 $ 
$$ 2\arcsin\big( \sinh(x) \big) + \arccos\big( 2 - \cosh(2x) \big) = 0 $$

If this is true, then for what values of $x$?  (Clearly, the $\arcsin(\cdot)$ and $\arccos(\cdot)$ are not valid for every real argument.)

I suppose a typo; changing the $+$ to $-$ makes
$$2\arcsin\big( \sinh(x) \big) \color{red}{-} \arccos\big( 2 - \cosh(2x) \big) = 0$$  true for all $x \geq 0$.
Composing Taylor series built at $x=0$ and assuming $x >0$, we have
$$2\arcsin\big( \sinh(x) \big)=2 x+\frac{2 x^3}{3}+\frac{x^5}{3}+\frac{79 x^7}{315}+\frac{493
   x^9}{2268}+O\left(x^{11}\right)$$
$$\arccos\big( 2 - \cosh(2x) \big)=2 x+\frac{2 x^3}{3}+\frac{x^5}{3}+\frac{79 x^7}{315}+\frac{493
   x^9}{2268}+O\left(x^{11}\right)$$
