The principal value of \( \cos^{-1}\left(\displaystyle\frac{1}{2}\right)+\sin^{-1}\left(\displaystyle\frac{-1}{\sqrt{2}}\right) \) is :
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( \displaystyle\frac{\pi}{12} \)
The principal value branch of \( \cos^{-1} \) is \( [0,\pi] \) and that of \( \sin^{-1} \) is \( \left[-\frac{\pi}{2},\frac{\pi}{2}\right] \).
\( \displaystyle \cos^{-1}\left(\frac{1}{2}\right)=\frac{\pi}{3} \)
\( \displaystyle \sin^{-1}\left(-\frac{1}{\sqrt{2}}\right)=-\sin^{-1}\left(\frac{1}{\sqrt{2}}\right) \)
\( \displaystyle =-\frac{\pi}{4} \)
\( \displaystyle \frac{\pi}{3}-\frac{\pi}{4}=\frac{4\pi-3\pi}{12}=\frac{\pi}{12} \)
Tip: Use \( \sin^{-1}(-x)=-\sin^{-1}x \) but \( \cos^{-1}(-x)=\pi-\cos^{-1}x \); mixing up these two is the most common error in principal-value questions.