Proof in complex

Question

Prove that for each $z\in \mathbb{C}$ we have $$\overline{\cos z} = \cos \overline{z}$$ using $ \cos z = {e^{iz}+e^{-iz}\over 2}$.

Can you please give me a link or show this proff using the equation i wrote. My english sucks so i can really search for it.

You ought to to try and learn the basic steps in such a question: First, you are asked to prove $X=Y$, so start with $X$, then use all the mathematical tools you have available to you to obtain "Y". Then you're done. They even tell you how to do it (use the exponent form of $\cos$). So the only thing remaining is for you to know what the "conjugate" is ($\bar{z}$). You should be able to use this approach to get from $X$ to $Y$. — pshmath0, Jan 07 '18 at 11:24

score 4 · Accepted Answer · answered Jan 07 '18 at 11:19

\begin{eqnarray} \overline{\cos z} &=& \overline{e^{iz}+e^{-iz}\over 2}\\ &=& { \overline{e^{iz}}+\overline{e^{-iz}}\over 2}\\ &=& { e^{\overline{iz}}+{e^{\overline{-iz}}}\over 2}\\ &=& { e^{-i\overline{z}}+{e^{i\overline{z}}}\over 2}\\ &=& \cos{\overline{z}} \end{eqnarray}

score 0 · Answer 2 · answered Jan 07 '18 at 11:14

0

Hint: $$\overline{e^z} = e^{\overline{z}}.$$

answered Jan 07 '18 at 11:14

TheGeekGreek

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Proof in complex

2 Answers2