I'm having trouble with this assignment:
Show that $ \overline {\cos(z)} = \cos(\bar z)$, where $\bar z$ represents conjugate of $z$.
I know that the two are equal, but how to mathematically show it?
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$\begingroup$you can use Euler's formula to show that $\cos(ix)= \cosh(x)$ and $\sin(ix)= i\sinh(x)$ $$\begin{align*} \overline{\cos(x + iy)} &= \overline{ \cos(x) \cos(iy) - \sin(x)\sin(iy)}\\ &= \overline{\cos(x)\cosh(y)- i \sin(x)\sinh(y)}\\ &= \cos(x)\cosh(y)+ i \sin(x)\sinh(y)\\ &= \cos(x)\cos(iy)+ \sin(x)\sin(iy)\\ &= \cos(x - iy) \\ &= \cos( \overline{x+iy}) \end{align*}$$
$\endgroup$ 0 $\begingroup$Well i do it this way:
\begin{align*} \overline{\cos(z)} &= \frac {\overline {e^{iz} + e^{-iz}}}{\overline{2}} \\ &= \frac {\overline {e^{iz}} + \overline {e^{-iz}}}{2} \\ &= \frac {e^\overline{iz} + e^\overline{-iz}}{2} \\ &= \frac {e^{-i\overline{z}} + e^{i\overline{z}}}{2} \\ &= \frac {e^{i\overline{z}} + e^{-i\overline{z}}}{2} \\ &= cos(\overline{z}) \end{align*}
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