Is it true that for every $z \in \mathbb{C}$
$$\begin{align} &2\mathfrak{Re}(\sinh z) &= \sinh z + \sinh \bar z\\ &2i\cdot \mathfrak{Im}(\sinh z) &= \sinh z - \sinh \bar z \end{align}$$
Is it true that for every $z \in \mathbb{C}$
$$\begin{align} &2\mathfrak{Re}(\sinh z) &= \sinh z + \sinh \bar z\\ &2i\cdot \mathfrak{Im}(\sinh z) &= \sinh z - \sinh \bar z \end{align}$$
This is true because the coefficients of the power series expansion of these entire functions are real.