Does $\tan(x)$ and $\cot(x)$ has symmetry axis? (like e.g $\cos(x)$ at $\pi k$ for $k \in \mathbb{Z}$), I tried think in the direction that $\sin(x)/\cos(x) = \tan(x)$ and both of them have symmetry axis but I couldn't quite get around the solution.
[Math] $\tan(x), \cot(x)$ function properties
functionstrigonometry
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Best Answer
Geometric approach
Let's first look at the graphs of $\color{blue} \tan \color{blue}x$ and $\color{red} \cot \color{red} x$. (Note: the two graphs face each other in opposite directions)
Both tangent and cotangent have origin symmetry, and this means they are odd.
Algebraic approach
Foreknowledge: $\sin x$ is odd ($\sin(-x)=-\sin(x)$), $\cos x$ is even ($\cos(-x)=\cos(x)$).
$\tan(-x)=\frac{\sin(-x)}{\cos(-x)}=\frac{-\sin(x)}{\cos(x)}=-\tan(x)$
$\cot(-x)=\frac{1}{\tan(-x)}=\frac{\cos(-x)}{\sin(-x)}=\frac{\cos(x)}{-\sin(x)}=-\cot(x)$