Solving an equation with hyperbolic functions I'm trying to prove that for a given $s,t\in\mathbb{R}$ there exists $w\in\mathbb{R}$ such that 

$\cosh(t)e^{i(s+w)}+\sinh(t)e^{i(s-w)}\in\mathbb{R}$.

How to solve this?
The imaginary part must be zero.
$$\cosh(t)\sin(s+w)+\sinh(t)\sin(s-w)=0$$
$$\cosh(t)\sin(s)\cos(w)+\cosh(t)\cos(s)\sin(w)+\sinh(t)\sin(s)\cos(w)-\sinh(t)\cos(s)\sin(w)=0$$
$$\tan(w)=\frac{\sinh(t)\cos(s)-\cosh(t)\cos(s)}{\cosh(t)\sin(s)+\sinh(t)\sin(s)}=-e^{-2t}\cot(s).$$
Try this:

1) divide by $\sinh(t)$ by eliminating from the domain the $t$ values for which $\sinh(t) = 0$.

2) divide by $e^{i(s-w)}$

3) Arrange the exponential to obtain in the end

$$e^{2iw} + \tanh(t)$$

Now proceed.