Why does $|e^{ix}|^2 = 1$?

If $x \in \mathbb{R}$, $$\cos^2x - \sin^2 x= \cos(2x)$$

$$2 \sin x \cos x =\sin(2x)$$

$$|(\cos x + i \sin x)^2|=\cos^2(2x)+\sin^2(2x)=1$$

If $a,b$ are real then $\displaystyle \left| a+bi \right| = \sqrt{a^2+b^2\,\,} = \sqrt{(a+bi)(a-bi)\,}.$