Question:
Find the principal value of:
\[ \sin^{-1}\left\{\cos\left(\sin^{-1}\left(\frac{\sqrt{3}}{2}\right)\right)\right\} \]
Solution:
Step 1: Evaluate inner inverse
\[ \sin^{-1}\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{3} \]
Step 2: Substitute
\[ \sin^{-1}\left(\cos \frac{\pi}{3}\right) \]
Step 3: Evaluate cosine
\[ \cos \frac{\pi}{3} = \frac{1}{2} \]
Step 4: Final inverse
\[ \sin^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{6} \]
—Final Answer:
\[ \boxed{\frac{\pi}{6}} \]