Question
If \( x < 0 \), \( y < 0 \) and \( xy = 1 \), find:
\[ \tan^{-1}x + \tan^{-1}y \]
Solution
We use identity:
\[ \tan^{-1}x + \tan^{-1}y = \tan^{-1}\left(\frac{x+y}{1-xy}\right) \]
Given:
\[ xy = 1 \Rightarrow 1 – xy = 0 \]
So expression becomes:
\[ \tan^{-1}(\pm \infty) \]
Now determine sign:
\[ x < 0,\; y < 0 \Rightarrow x + y < 0 \]
Hence,
\[ \tan^{-1}(-\infty) = -\frac{\pi}{2} \]
Final Answer:
\[ \boxed{-\frac{\pi}{2}} \]
Key Concept
When denominator becomes zero, check sign of numerator to determine ±π/2.