Problem
Evaluate: \( \cos\left(\sec^{-1}x + \csc^{-1}x\right), \quad |x| \ge 1 \)
Solution
Let:
\[ A = \sec^{-1}x,\quad B = \csc^{-1}x \]
Step 1: Convert to sine and cosine
\[ \sec A = x \Rightarrow \cos A = \frac{1}{x} \]
\[ \csc B = x \Rightarrow \sin B = \frac{1}{x} \]
Step 2: Key Identity Insight
\[ \sin B = \cos A \Rightarrow B = \frac{\pi}{2} – A \]
\[ A + B = \frac{\pi}{2} \]
Step 3: Evaluate cosine
\[ \cos\left(A + B\right) = \cos\left(\frac{\pi}{2}\right) \]
\[ = 0 \]
Final Answer
\[ \boxed{0} \]
Explanation
Since sec⁻¹x + cosec⁻¹x = π/2 for |x| ≥ 1, the cosine becomes zero.