Match List I with List II
| LIST I | LIST II | ||
| A. | \(\sin^{-1}x+\cos^{-1}x, x \in [-1,1]\) | I. | \(-\displaystyle\frac{\pi}{2}\) |
| B. | \(\tan^{-1}\sqrt{3}-\cot^{-1}\left(-\sqrt{3}\right)\) | II. | \(-\displaystyle\frac{\pi}{6}\) |
| C. | \(\cos^{-1}\left(\cos\displaystyle\frac{13\pi}{6}\right)\) | III. | \(\displaystyle\frac{\pi}{2}\) |
| D. | \(\sin^{-1}\left(-\displaystyle\frac{1}{2}\right)\) | IV. | \(\displaystyle\frac{\pi}{6}\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → A-III, B-I, C-IV, D-II
(A) \(\sin^{-1}x+\cos^{-1}x, x \in [-1,1]\)
This matches (III) \(\frac{\pi}{2}\).
This is the standard identity \(\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}\) for every x in [−1, 1].
(B) \(\tan^{-1}\sqrt{3}-\cot^{-1}\left(-\sqrt{3}\right)\)
This matches (I) \(-\frac{\pi}{2}\).
The principal value of \(\cot^{-1}\) lies in \((0, \pi)\), so \(\cot^{-1}(-x) = \pi - \cot^{-1}x\):
\( \displaystyle \tan^{-1}\sqrt{3} = \frac{\pi}{3} \)
\( \displaystyle \cot^{-1}\left(-\sqrt{3}\right) = \pi - \frac{\pi}{6} = \frac{5\pi}{6} \)
\( \displaystyle \frac{\pi}{3} - \frac{5\pi}{6} = -\frac{\pi}{2} \)
(C) \(\cos^{-1}\left(\cos\frac{13\pi}{6}\right)\)
This matches (IV) \(\frac{\pi}{6}\).
\(\frac{13\pi}{6}\) is not in the principal range \([0, \pi]\), so first reduce it:
\( \displaystyle \cos\frac{13\pi}{6} = \cos\left(2\pi + \frac{\pi}{6}\right) = \cos\frac{\pi}{6} \)
\( \displaystyle \cos^{-1}\left(\cos\frac{\pi}{6}\right) = \frac{\pi}{6} \)
(D) \(\sin^{-1}\left(-\frac{1}{2}\right)\)
This matches (II) \(-\frac{\pi}{6}\).
The principal value of \(\sin^{-1}\) lies in \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\), and \(\sin\left(-\frac{\pi}{6}\right) = -\frac{1}{2}\), so \(\sin^{-1}\left(-\frac{1}{2}\right) = -\frac{\pi}{6}\).
Tip: For a negative argument, \(\sin^{-1}(-x) = -\sin^{-1}x\) and \(\tan^{-1}(-x) = -\tan^{-1}x\), but \(\cos^{-1}(-x) = \pi - \cos^{-1}x\) and \(\cot^{-1}(-x) = \pi - \cot^{-1}x\). Taking \(\cot^{-1}(-\sqrt{3}) = -\frac{\pi}{6}\) is a common slip.