We have to prove that the tangent is an odd function.
If the tangent is an odd function, the following condition should be satisfied:
[tex]\tan(t)=-\tan(-t)[/tex]
From the figure we can see that the tangent can be expressed as:
We can start then from tan(t) and will try to arrive to -tan(-t):
[tex]\begin{gathered} \tan(t)=\frac{\sin(t)}{cos(t)}=\frac{y}{x} \\ \tan(t)=\frac{-(-y)}{x}=\frac{-\sin(-t)}{\cos(-t)} \\ \tan(t)=-\frac{\sin(-t)}{\cos(-t)} \\ \tan(t)=-\tan(-t) \end{gathered}[/tex]
We have arrived to the condition for odd functions, so we have just proved that the tangent function is an odd function.
