(related to Proposition: Eveness of the Cosine of a Real Variable)
Following the representation of the real cosine by the complex exponential function, the rules of multiplying positive and negative real numbers, the commutativity of multiplying and the commutativity of adding complex numbers, we get
\[\begin{array}{rcll} \cos(-x)&=&\frac 12(\exp(i(-x))+\exp(-i(-x))&\text{representing of cosine by complex exponential function}\\ &=&\frac 12(\exp((-x) i)+\exp((-1)(-x)i)&\text{commutativity of multiplying complex numbers}\\ &=&\frac 12(\exp(-ix)+\exp(ix))&\text{multiplying positive and negative real numbers}\\ &=&\frac 12(\exp(ix)+\exp(-ix))&\text{commutativity of adding complex numbers}\\ &=&\cos(x)&\text{representing of cosine by complex exponential function} \end{array}\]
Thus, the cosine of a real variable is an even function.
