Match List-I with List-II

Where $\mathbb{R}$ is set of real numbers

List-I	List-II
(A) $\sin x$ is continuous on:	(I) $\mathbb{R} - \{0\}$
(B) $\tan x$ is continuous on:	(II) $\mathbb{R}$
(C) $\cot x$ is continuous on:	(III) $\mathbb{R} - \{n\pi: n \in \mathbb{Z}\}$
(D) $x^{-n}, n \in \mathbb{N}$ is continuous on:	(IV) $\mathbb{R} - \left\{(2n + 1)\frac{\pi}{2}: n \in \mathbb{Z}\right\}$

Choose the correct answer from the options given below:

A function is continuous on a set if there's no break, jump, or hole in its graph at every point in that set. A function is not continuous where it's undefined.

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$\sin x$ is defined for all real numbers. The graph is a smooth wave with no breaks anywhere.

$\sin x$ is continuous on $\mathbb{R}$

(A) matches with (II)

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$\tan x = \frac{\sin x}{\cos x}$

$\tan x$ is undefined when $\cos x = 0$

$\cos x = 0$ at $x = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, -\frac{\pi}{2}, -\frac{3\pi}{2}, ...$

General form: $x = (2n + 1)\frac{\pi}{2}$ where $n \in \mathbb{Z}$

$\tan x$ is continuous on $\mathbb{R} - \left\{(2n + 1)\frac{\pi}{2}: n \in \mathbb{Z}\right\}$

(B) matches with (IV)

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$\cot x = \frac{\cos x}{\sin x}$

$\cot x$ is undefined when $\sin x = 0$

$\sin x = 0$ at $x = 0, \pi, 2\pi, 3\pi, -\pi, -2\pi, ...$

General form: $x = n\pi$ where $n \in \mathbb{Z}$

$\cot x$ is continuous on $\mathbb{R} - \{n\pi: n \in \mathbb{Z}\}$

(C) matches with (III)

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$x^{-n} = \frac{1}{x^n}$ where $n \in \mathbb{N}$

This is undefined when $x = 0$

$x^{-n}$ is continuous on $\mathbb{R} - \{0\}$

(D) matches with (I)

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Final matching:

(A) $\sin x$ → (II) $\mathbb{R}$

(B) $\tan x$ → (IV) $\mathbb{R} - \left\{(2n + 1)\frac{\pi}{2}: n \in \mathbb{Z}\right\}$

(C) $\cot x$ → (III) $\mathbb{R} - \{n\pi: n \in \mathbb{Z}\}$

(D) $x^{-n}$ → (I) $\mathbb{R} - \{0\}$