Cosine Function is Even

Theorem

Let $x \in \R$ be a real number.

Let $\cos x$ be the cosine of $x$.

Then:

$\cos \left({-x}\right) = \cos x$

That is, the cosine function is even.

Proof

Recall the definition of the cosine function:

$\displaystyle \cos x = \sum_{n=0}^\infty \left({-1}\right)^n \frac {x^{2n}}{\left({2n}\right)!} = 1 - \frac {x^2} {2!} + \frac {x^4} {4!} - \cdots$

From Even Powers are Positive, we have that:

$\forall n \in \N: x^{2n} = \left({-x}\right)^{2n}$

The result follows.

$\blacksquare$

Also see

• Sine Function is Odd
• Tangent Function is Odd
• Hyperbolic Cosine Function is Even

Sources

• Murray R. Spiegel: Mathematical Handbook of Formulas and Tables (1968): $5.29$
• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 16.3 \ (1) \ \text{(iii)}$