Cosine Function is Even/Proof 1
Theorem
- $\map \cos {-z} = \cos z$
That is, the cosine function is even.
Proof
Recall the definition of the cosine function:
| \(\ds \cos z\) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {z^{2 n} } {\paren {2 n}!}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 1 - \frac {z^2} {2!} + \frac {z^4} {4!} - \cdots\) |
From Even Power is Non-Negative:
- $\forall n \in \N: z^{2 n} = \paren {-z}^{2 n}$
The result follows.
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.5$. The Functions $e^z$, $\cos z$, $\sin z$: $\text{(ii)}$: $(4.15)$
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 16.3 \ (1) \ \text{(iii)}$