Cosine Exponential Formulation/Real Domain/Proof 4

Theorem
For any complex number $x$:
 * $\cos x = \dfrac {e^{i x} + e^{-i x} } 2$

where $e^{ix}$ is the complex exponential function, $\cos$ is cosine and $i$ is the imaginary unit.

Proof
Consider the differential equation:
 * $D^2_x f \left({x}\right) = - f\left({x}\right)$


 * $f \left({0}\right) = 1$


 * $D_x f \left({0}\right) = 0$

Step 1
We will prove that $y = \cos x$ is a solution.

Step 2
We will prove that $z = \dfrac {e^{ix}+e^{-ix}} 2$ is a solution.

So $y$ and $z$ are both specific solutions.

But a specific solution to a differential equation is unique.

Therefore, $y=z$, i.e. $\cos x = \dfrac {e^{ix} + e^{-ix} } 2$