A calculus of the absurd

14.7.3 Using the exponential form to show odd/evenness

Theorem 14.7.1 The cosine function is even, that is
\(\seteqnumber{0}{14.}{70}\)
\begin{equation} \cos (x) = \cos (-x) \end{equation}

Proof:

\begin{align} \cos (x) &= \frac {e^{ix} + e^{-ix}}{2} \\ &= \frac {e^{-ix} + e^{ix}}{2} & \text {As addition is commutative} \\ &= \frac {e^{i(-x)} + e^{-i(-x)}}{2} \\ &= \cos (-x) \end{align}

Theorem 14.7.2 The sine function is odd, that is
\(\seteqnumber{0}{14.}{75}\)
\begin{equation} \sin (x) = -\sin (-x) \end{equation}

Proof:

\begin{align} \sin (x) &= \frac {e^{ix} - e^{-ix}}{-2i} \\ &= - \left (\frac {e^{-ix} - e^{ix}}{-2i}\right ) \\ &= - \left (\frac {e^{i(-x)} - e^{-i(-x)}}{-2i}\right ) \\ &= - \sin (-x) \end{align}