Match List – I with List – II.
| LIST I | LIST II | ||
| A. | Range of \(\mathrm{cosec}^{-1}x\) | I. | \((0, \pi)\) |
| B. | Range of \(\tan^{-1}x\) | II. | \([0, \pi]-\left\{\displaystyle\frac{\pi}{2}\right\}\) |
| C. | Range of \(\sec^{-1}x\) | III. | \(\left[-\displaystyle\frac{\pi}{2}, \frac{\pi}{2}\right]-\{0\}\) |
| D. | Range of \(\cot^{-1}x\) | IV. | \(\left(-\displaystyle\frac{\pi}{2}, \frac{\pi}{2}\right)\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → A-III, B-IV, C-II, D-I
Each range is the principal value branch of the function.
(A) Range of \(\mathrm{cosec}^{-1}x\)
This matches (III) \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]-\{0\}\). The branch is that of \(\sin^{-1}\), with 0 removed because cosec is never defined at 0.
(B) Range of \(\tan^{-1}x\)
This matches (IV) \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\). The end points are excluded because tan is not defined at \(\pm\frac{\pi}{2}\).
(C) Range of \(\sec^{-1}x\)
This matches (II) \([0, \pi]-\left\{\frac{\pi}{2}\right\}\). The branch is that of \(\cos^{-1}\), with \(\frac{\pi}{2}\) removed because sec is not defined there.
(D) Range of \(\cot^{-1}x\)
This matches (I) \((0, \pi)\). The end points are excluded because cot is not defined at 0 and \(\pi\).
NCERT (2026-27 edition):
Basic Concepts
