Prove that the sine of a complex conjugate is equal to the conjugate of the sine of the complex number

Question

I need to prove that given $z\in \mathbb{C}$ it follows that:

$\sin(\bar z)= \overline{\sin(z)}$

I don't know how to start the proof, so any advice can be useful. Thanks!

Answer 1

Hint

You can use the Taylor expansion of $$\sin(z)= z-z^3/3!+ z^5/5!+\dots$$

Also use the property of conjugate that $$\overline{z_1 + z_2}=\overline{z_1} + \overline{z_2}$$

$$\overline{z_1 \times z_2}=\overline{z_1} \times \overline{z_2}$$