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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

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

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