Question
Evaluate:
\[ \cos\left(\frac{\tan^{-1}x + \cot^{-1}x}{3}\right) \quad \text{when } x = -\frac{1}{\sqrt{3}} \]
Solution
We use identity:
\[ \tan^{-1}x + \cot^{-1}x = \frac{\pi}{2} \quad \text{for all } x \in \mathbb{R} \]
So,
\[ \cos\left(\frac{\pi/2}{3}\right) = \cos\left(\frac{\pi}{6}\right) \]
\[ = \frac{\sqrt{3}}{2} \]
Final Answer:
\[ \boxed{\frac{\sqrt{3}}{2}} \]
Key Concept
The identity \( \tan^{-1}x + \cot^{-1}x = \frac{\pi}{2} \) simplifies such expressions instantly.