Question
Express:
\[ \cot^{-1}(-x) \]
in terms of \( \cot^{-1}x \), where \( x \in \mathbb{R} \).
Solution
Let
\[ \cot^{-1}x = \theta \]
Then,
\[ x = \cot \theta \]
So,
\[ -x = -\cot \theta = \cot(\pi – \theta) \]
Thus,
\[ \cot^{-1}(-x) = \pi – \theta \]
Substitute back \( \theta = \cot^{-1}x \):
\[ \cot^{-1}(-x) = \pi – \cot^{-1}x \]
Final Answer:
\[ \boxed{ \cot^{-1}(-x) = \pi – \cot^{-1}x } \]
Key Concept
Use the identity \( \cot(\pi – \theta) = -\cot \theta \) and principal value range \( (0, \pi) \).