which of the following are the same for both the tangent and cotangent functions? domain range asymptotes…

Brief Explanations:

1. Domain: The domain of $y = \tan(x)$ is all real - numbers except $x=\frac{\pi}{2}+k\pi,k\in\mathbb{Z}$, and the domain of $y=\cot(x)$ is all real - numbers except $x = k\pi,k\in\mathbb{Z}$. So, the domains are different.
2. Range: The range of $y = \tan(x)$ and $y=\cot(x)$ is $(-\infty,\infty)$.
3. Asymptotes: The vertical asymptotes of $y=\tan(x)$ are $x=\frac{\pi}{2}+k\pi,k\in\mathbb{Z}$, and for $y = \cot(x)$ are $x=k\pi,k\in\mathbb{Z}$. So, the asymptotes are different.
4. Period: The period of $y=\tan(x)$ is $\pi$, and the period of $y=\cot(x)$ is also $\pi$.
5. Intercepts: The $x$ - intercepts of $y=\tan(x)$ are $x = k\pi,k\in\mathbb{Z}$, and for $y=\cot(x)$ there are no $x$ - intercepts (since $\cot(x)=\frac{\cos(x)}{\sin(x)}$ and is zero when $\cos(x) = 0$ but is undefined at those points). So, the intercepts are different.

Answer:

range, period