Does this equivalence stand?

Question

I'm struggling to verify the following equivalence from Ekeland (p.12):

$$\lambda (\xi \cosh t + \overline \xi \sinh t) = \lambda e^t (\xi + \overline \xi)$$

for $\lambda=\pm 1$ and $\xi \in \mathbb C^n$

Answer 1

Let $\xi=i$, $\lambda=1$: $$ \xi\cosh(t)+\bar{\xi}\sinh(t)=i(\cosh(t)-\sinh(t))=ie^{-t}\neq 0 $$

Hence the relation is false.