Write the principal value

Question:

Write the principal value of $\sin ^{-1}\left\{\cos \left(\sin ^{-1} \frac{1}{2}\right)\right\}$

Solution:

$\sin ^{-1}\left\{\cos \left(\sin ^{-1} \frac{1}{2}\right)\right\}=\sin ^{-1}\left\{\cos \left[\sin ^{-1}\left(\sin \frac{\pi}{3}\right)\right]\right\}$

$=\sin ^{-1}\left[\cos \left(\frac{\pi}{3}\right)\right]$

$=\sin ^{-1}\left[\frac{1}{2}\right]$

$=\sin ^{-1}\left[\sin \left(\frac{\pi}{3}\right)\right]$

$=\frac{\pi}{3}$

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