Find all points at which tangent and cotangent are continuous

Find the points $x \in \mathbb{R}$ at which $\tan x$ is continuous and at which $\cot x$ is continuous.

Since

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

and $\sin x$ , $\cos x$ are continuous everywhere we know (by Theorem 3.2) that $\tan x$ is continuous everywhere $\cos x$ is not zero. We proved in this exercise (Section 2.8, #1 (b) of Apostol) that

$\cos x = 0 \quad \iff \quad x = \frac{\pi}{2} + n \pi, \qquad n \in \mathbb{Z}.$

Thus, $\tan x$ is continuous for

$\left\{ x \in \mathbb{R} \mid x \neq \frac{\pi}{2} + n \pi, \ n \in \mathbb{Z} \right\}$

Similarly,

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

and by Theorem 3.2 we then have $\cot x$ is continuous everywhere that $\sin x$ is not zero. We proved in the same exercise (Section 2.8, #1(a) of Apostol) that

$\sin x = 0 \quad \iff \quad x = n \pi \qquad n \in \mathbb{Z}.$

Hence, $\cot x$ is continuous for

$\left\{ x \in \mathbb{R} \mid x \neq n \pi, n \in \mathbb{Z} \right \}.$

Stumbling Robot

Find all points at which tangent and cotangent are continuous

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