Modulus of Exponential of Imaginary Number is One

Theorem
Let $\left|{z}\right|$ be the complex modulus of a complex number $z$.

Let $e^z$ be the complex exponential of $z$.

Let $x$ be wholly real.

Then:


 * $\left \vert {e^{ix}}\right \vert = 1$.

Corollary
Let $t > 0$ be wholly real.

Let $t^{i x}$ be $t$ to the power of $ix$ defined on its principal branch.

Then:


 * $\left \vert {t^{ix}}\right \vert = 1$.