The range of the principal value branch of \(\tan^{-1} x\) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\left(-\displaystyle\frac{\pi}{2}, \displaystyle\frac{\pi}{2}\right)\)
On \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\) the tangent function is one-one and takes every real value, so this interval is the principal value branch of \(\tan^{-1}\).
The end points are not included, because \(\tan\frac{\pi}{2}\) and \(\tan\left(-\frac{\pi}{2}\right)\) are not defined. As x grows, \(\tan^{-1} x\) comes closer and closer to \(\frac{\pi}{2}\) but never reaches it.
Why the other options are incorrect
Option (2) \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) — Incorrect. This is the range of \(\sin^{-1}\); for \(\tan^{-1}\) the end points are excluded.
Option (3) \((0, \pi)\) — Incorrect. This is the range of \(\cot^{-1}\).
Option (4) \([0, \pi] - \left\{\frac{\pi}{2}\right\}\) — Incorrect. This is the range of \(\sec^{-1}\).