Convert $1 + e^{2i}$ to $e^i \cos(1)$

Question

Reading a long solution I saw this a step that converts $1 + e^{2i}$ to $e^i \cos(1)$. How is this done?

How do I generalize this?

Answer 1

There's a factor of $2$ missing.

Generally,

$$1 + e^{2i\varphi} = e^{i\varphi}(e^{-i\varphi} + e^{i\varphi}) = e^{i\varphi}\cdot 2\cos \varphi$$

since

$$\cos \varphi = \frac{e^{i\varphi} + e^{-i\varphi}}{2}.$$

Answer 2

Using Euler's formula we get:

$$1 + e^{2i} = 1 + e^{i} \cdot e^{i} = 1 + e^i \cdot i \sin(1) + e^i \cdot \cos(1)$$

With this approach you can see, that the difference between the two expression given above is $1 + e^i \cdot i \sin(1)$.

Answer 3

\begin{align}1+e^{2i}&=e^i(e^{-i}+e^i)\\&=e^i((\cos1-i\sin1)+(\cos1+i\sin1))&\\&=2e^i\cos1\end{align}