Evaluate: sec-1(√2) + 2cosec-1(−√2)
Solution:
Step 1: Evaluate sec⁻¹(√2)
\[ \sec^{-1}(\sqrt{2}) \Rightarrow \cos y = \frac{1}{\sqrt{2}} \]
\[ y = \frac{\pi}{4} \]
(Principal range: \([0,\pi], y \ne \pi/2\))
Step 2: Evaluate cosec⁻¹(−√2)
\[ \csc^{-1}(-\sqrt{2}) \Rightarrow \sin y = -\frac{1}{\sqrt{2}} \]
\[ y = -\frac{\pi}{4} \]
(Principal range: \([-\pi/2,0) \cup (0,\pi/2]\))
Step 3: Substitute
\[ \frac{\pi}{4} + 2\left(-\frac{\pi}{4}\right) \]
\[ = \frac{\pi}{4} – \frac{\pi}{2} = -\frac{\pi}{4} \]
Final Answer:
Value = \[ -\frac{\pi}{4} \]