(related to Lemma: Unit Circle)
In the following proof, we will use the following results: * definition of absolute value of complex numbers, * taking roots is inverse to taking squares, * complex conjugate of complex exponential function, * definition of the complex conjugate, * function equation of the complex exponential function, * the complex case of the result \(\exp(0)=1\), * existence of inverse elements with respect to addition of complex numbers, * definition of the real part:
\[\begin{array}{rcll} |\exp(ix)|^2&=&\left(\sqrt{\Re(\exp(ix)\cdot(\exp(ix))^*)}\right)^2&\text{definition of absolute value for complex numbers}\\ &=&\Re(\exp(ix)\cdot(\exp(ix))^*)&\text{taking roots is inverse to taking squares}\\ &=&\Re(\exp(ix)\cdot\exp((ix)^*))&\text{complex conjugate of complex exponential function}\\ &=&\Re(\exp(ix)\cdot\exp(-ix))&\text{definition of complex conjugate}\\ &=&\Re(\exp(ix+(-ix)))&\text{functional equation of}\exp\\ &=&\Re(\exp(0))&\text{existence of inverse elements with respect to addition of complex numbers}\\ &=&\Re(1)&\text{because}\exp(0))=1\\ &=&1&\text{definition of real part}\\ \end{array}\]
Taking square roots on both sides of the equation gives the desired result \[|\exp(ix)|=1.\]