Is the function $f(x) = \tan(x)$ odd, even, or neither?
Here is what I have so far:
I know the function is not even because $f(x) ≠ f(-x):$
$$f(-x) = \tan(-x)$$
$$\tan(-x) ≠ \tan(x)$$
Now I want to determine if the function is odd. I know a function is odd if $f(x) = -f(-x)$:
$$-f(-x) = -\tan(-x)$$
How does the negative sign on the outside of the brackets affect $\tan(-x)?$
My instinct is that the function is neither even nor odd, but I would like confirmation.
Best Answer
Just use the definition of $\tan$ in terms of $\sin$ and $\cos$, and use the fact that $\sin$ is odd, and $\cos$ is even :
$$\tan(-x)=\dfrac{\sin(-x)}{\cos(-x)}=\dfrac{-\sin(x)}{\cos(x)}=-\tan(x).$$