Domain of $f(x)=\frac{\operatorname{sgn}(x)}{\tan(x)}$

Question

I have to evaluate the domain of

$$f(x)=\displaystyle\frac{\mbox{sgn}(x)}{\tan(x)}$$

where $\mbox{sgn}(x)$ is the signum function.

The condition that define the domain is

$$\tan(x)\ne 0 \implies \frac{\sin(x)}{\cos(x)}\ne 0 \implies \sin(x)\ne 0\ \vee \ \cos(x)\ne 0$$

so the domain must be

$\mbox{Dom}(f)=\mathbb{R}-\left\{k\cdot\dfrac{\pi}{2}\right\}\ \ \mbox{with} \ k\in\mathbb{Z}$

but Wolfram doesn't agree with my solution. I need to understand where is my mistake. Thanks.

Answer 1

Hint

$$\tan(x)=0 \implies x= k\pi$$

and doesn't exist tangent at the point $\frac \pi 2 + k \pi$