Find the Domain of f(x) = cot x + cot-1(x)
Solution:
Given function:
\[ f(x) = \cot x + \cot^{-1}(x) \]
Step 1: Domain of cot x
\[ \cot x = \frac{\cos x}{\sin x} \]
So,
\[ \sin x \neq 0 \Rightarrow x \neq n\pi,\; n \in \mathbb{Z} \]
Step 2: Domain of cot⁻¹(x)
\[ \cot^{-1}(x) \text{ is defined for all real } x \]
Step 3: Intersection of domains
\[ \mathbb{R} \setminus \{n\pi,\; n \in \mathbb{Z}\} \]
Final Answer:
Domain = \[ \mathbb{R} – \{n\pi : n \in \mathbb{Z}\} \]