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:

So $x$ and $y$ have the same Laplace transform for $\theta \ge 0$.

Now define $\tau = -\theta, \sigma = -s$, and consider $\theta < 0$ so that $\tau > 0$.

Taking Laplace transforms of $x\left({\tau}\right)$ and $y\left({\tau}\right)$:

So $x\left({\theta}\right)$ and $y\left({\theta}\right)$ have the same Laplace transform for $\theta < 0$.

The result follows from Injectivity of Laplace Transform.