Cosine Function is Even

Theorem

For all $z \in \C$:

$\map \cos {-z} = \cos z$

That is, the cosine function is even.

Recall the definition of the cosine function:

$\ds \cos z$

$=$

$\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {z^{2 n} } {\paren {2 n}!}$

$\ds $

$=$

$\ds 1 - \frac {z^2} {2!} + \frac {z^4} {4!} - \cdots$

$\forall n \in \N: z^{2 n} = \paren {-z}^{2 n}$

The result follows.

$\blacksquare$

$\ds \map \cos {-z}$

$=$

$\ds \frac {e^{i \paren {-z} } + e^{-i \paren {-z} } } 2$

$\ds $

$=$

$\ds \frac {e^{i z} + e^{-i z} } 2$

simplifying

$\ds $

$=$

$\ds \cos z$

$\blacksquare$

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.29$
1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: Functions of Angles in All Quadrants in terms of those in Quadrant I
1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $2$: Functions, Limits and Continuity: The Elementary Functions: $4$
1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): cosine
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): trigonometric function
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): trigonometric function
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $12$: Trigonometric formulae: Symmetry
2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $14$: Trigonometric formulae: Symmetry