Tangent Function is Odd

Theorem

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

Let $\tan x$ be the tangent of $x$.

Then, whenever $\tan x$ is defined:

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

That is, the tangent function is odd.

Proof

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \tan \left({-x}\right)\)	\(=\)	\(\displaystyle \)	\(\displaystyle \frac {\sin \left({-x}\right)} {\cos \left({-x}\right)}\)	\(\displaystyle \)	\(\displaystyle \)	Definition of tangent
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \)	\(\displaystyle \frac {- \sin x} {\cos x}\)	\(\displaystyle \)	\(\displaystyle \)	Sine Function is Odd; Cosine Function is Even
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \)	\(\displaystyle - \tan x\)	\(\displaystyle \)	\(\displaystyle \)	Definition of tangent

$\blacksquare$

Also see

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