# How do you prove that tangent is an odd function?

Jan 30, 2015

A function is even if:

$f \left(- x\right) = f \left(x\right)$.

A function is odd if:

$f \left(- x\right) = - f \left(x\right)$.

In this case:
$y = \tan \left(- x\right) = \sin \frac{- x}{\cos} \left(- x\right) = \frac{- \sin \left(x\right)}{\cos} x = - \sin \frac{x}{\cos} x = - \tan \left(x\right)$,

for the simmetry of sinus and cosinus.