Question:
Find the principal value of:
\[ \sin^{-1}\left(\frac{1}{2}\right) – 2\sin^{-1}\left(\frac{1}{\sqrt{2}}\right) \]
Solution:
Step 1: Use standard values
\[ \sin^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{6} \]
\[ \sin^{-1}\left(\frac{1}{\sqrt{2}}\right) = \frac{\pi}{4} \]
Step 2: Substitute
\[ \frac{\pi}{6} – 2\left(\frac{\pi}{4}\right) \]
Step 3: Simplify
\[ = \frac{\pi}{6} – \frac{\pi}{2} \]
\[ = \frac{\pi – 3\pi}{6} = -\frac{2\pi}{6} = -\frac{\pi}{3} \]
—Final Answer:
\[ \boxed{-\frac{\pi}{3}} \]