cos(z)=2

$$\begin{eqnarray} \cos\left(z\right)&=&2\;\cdots\;z\in\mathbb{C} \end{eqnarray}$$

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$$\begin{eqnarray} \cos\left(z\right)&=&\frac{ e^{iz}+e^{-iz} }{2} \\&=&\frac{ X+X^{-1} }{2}\;\cdots\;X=e^{iz} \\\frac{ X+X^{-1} }{2}&=&2 \\X+X^{-1}&=&2\cdot2 \\X+X^{-1}&=&4 \\X-4+X^{-1}&=&0 \\X\cdot\left(X-4-X^{-1}\right)&=&X\cdot0 \\X^2-4X+1&=&0 \\X&=&\frac{-(-4)\pm\sqrt{(-4)^2-4\cdot1\cdot 1}}{2\cdot1} \\&=&\frac{4\pm\sqrt{12}}{2} \\&=&\frac{4\pm2\sqrt{3}}{2} \\&=&2 \pm \sqrt{3} \\X=e^{iz}&=&2 \pm \sqrt{3} \\iz&=&\log{\left(2\pm \sqrt{3}\right)}+2\pi i n \;\cdots\; n \in \mathbb{Z} \\z&=&\frac{1}{i}\left\{\log{\left(2\pm \sqrt{3}\right)}+2\pi i n\right\} \\&=&\frac{i}{i}\frac{1}{i}\log{\left(2\pm \sqrt{3}\right)}+\frac{1}{\cancel{i}}2\pi \cancel{i} n \\&=&\frac{i}{-1}\log{\left(2\pm \sqrt{3}\right)}+2\pi n \\&=&-i\log{\left(2\pm \sqrt{3}\right)}+2\pi n \end{eqnarray}$$

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$$\begin{eqnarray} \frac{ e^{iz}+e^{-iz} }{2} &=& \frac{ e^{iz}+e^{i(-z)} }{2} \\&=& \frac{ \left(\cos{\left(z\right)}+i\sin{\left(z\right)}\right) +\left(\cos{\left(-z\right)}+i\sin{\left(-z\right)}\right) }{2} \\&=& \frac{ \left(\cos{\left(z\right)}+i\sin{\left(z\right)}\right) +\left(\cos{\left(z\right)}-i\sin{\left(z\right)}\right) }{2} \\&=& \frac{ \cos{\left(z\right)}+\cancel{i\sin{\left(z\right)}} +\cos{\left(z\right)}\cancel{-i\sin{\left(z\right)}} }{2} \\&=& \frac{ \cancel{2}\cos{\left(z\right)} }{\cancel{2}} \\&=&\cos{\left(z\right)} \end{eqnarray}$$