Evaluate: sin-1[cos{2cosec-1(−2)}]
Solution:
Step 1: Evaluate cosec⁻¹(−2)
\[ \csc^{-1}(-2) \Rightarrow \sin y = -\frac{1}{2} \]
\[ y = -\frac{\pi}{6} \]
(Principal range: \([-\pi/2,0) \cup (0,\pi/2]\))
Step 2: Multiply by 2
\[ 2y = 2\left(-\frac{\pi}{6}\right) = -\frac{\pi}{3} \]
Step 3: Evaluate cosine
\[ \cos\left(-\frac{\pi}{3}\right) = \frac{1}{2} \]
Step 4: Apply sin⁻¹
\[ \sin^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{6} \]
(Principal range: \([- \pi/2, \pi/2]\))
Final Answer:
Value = \[ \frac{\pi}{6} \]