Euler's Formula/Real Domain/Proof 4

Proof
Note that the following proof, as written, only holds for real $\theta$.

Define: $x \left({\theta}\right) = e^{i \theta}$ $y \left({\theta}\right) = \cos \theta + i \sin \theta$

Consider first $\theta \ge 0$.

Taking Laplace transforms:

Define $\tau = -\theta, \sigma = -s$.

Now consider $\theta < 0 \implies \tau > 0$, and take Reverse Laplace transforms:

By the same reasoning as above, $x$ and $y$ will have the same Laplace transforms, for $\tau > 0 \implies \theta < 0$.

The result follows from Uniqueness of Laplace Transform:Corollary.