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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?

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

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you can use this $$ \sinh(t) = \frac{e^{t} - e^{-t}}{2} \qquad \text{and} \qquad \cosh(t) = \frac{e^{t} + e^{-t}}{2}$$ to replace in your equation and then use the Euler's form to obtain a Complex number. This new number is in $\mathbb{C}$ and to this is only in $\mathbb{R}$, its imaginary part should be zero. Then you only resolve this new equation and finish your proof.

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