Tangent Function is Odd

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Theorem

For all $x \in \C$ where $\tan x$ is defined:

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

That is, the tangent function is odd.


Proof

\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \tan \left({-x}\right)\) \(=\) \(\displaystyle \) \(\displaystyle \frac {\sin \left({-x}\right)} {\cos \left({-x}\right)}\) \(\displaystyle \) \(\displaystyle \)          Tangent is Sine divided by Cosine          
\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(=\) \(\displaystyle \) \(\displaystyle \frac {- \sin x} {\cos x}\) \(\displaystyle \) \(\displaystyle \)          Sine Function is Odd; Cosine Function is Even          
\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(=\) \(\displaystyle \) \(\displaystyle - \tan x\) \(\displaystyle \) \(\displaystyle \)          Tangent is Sine divided by Cosine          

$\blacksquare$


Also see


Sources