Tangent Function is Odd

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Theorem

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

$\map \tan {-x} = -\tan x$

That is, the tangent function is odd.


Proof

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

$\blacksquare$


Also see


Sources

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