Consider the following statements about inverse trigonometric functions:
(A) The domain of \(\tan^{-1} x\) is the set of all real numbers.
(B) The domain of \(\cos^{-1} x\) is \([0, \pi]\).
(C) The range of the principal value branch of \(\sec^{-1} x\) is \([0, \pi] - \left\{\displaystyle\frac{\pi}{2}\right\}\).
(D) The domain of \(\cot^{-1} x\) is \((0, \pi)\).
Which of the above statements are correct?
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) and (C) only
The domain of an inverse trigonometric function is the set of values the trigonometric function takes, and its range is the principal value branch:
\(\tan^{-1}\): domain \(\mathrm{R}\), range \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\)
\(\cos^{-1}\): domain \([-1, 1]\), range \([0, \pi]\)
\(\sec^{-1}\): domain \(\mathrm{R} - (-1, 1)\), range \([0, \pi] - \left\{\frac{\pi}{2}\right\}\)
\(\cot^{-1}\): domain \(\mathrm{R}\), range \((0, \pi)\)
So the domains are \(\mathrm{R}\), \([-1, 1]\), \(\mathrm{R} - (-1, 1)\), \(\mathrm{R}\), and the range of \(\sec^{-1}\) is \([0, \pi] - \left\{\frac{\pi}{2}\right\}\).
Checking the statements
(A) The domain of \(\tan^{-1} x\) is the set of all real numbers. — Correct. Tangent takes every real value.
(B) The domain of \(\cos^{-1} x\) is \([0, \pi]\). — Incorrect. The domain is \([-1, 1]\); \([0, \pi]\) is its range.
(C) The range of the principal value branch of \(\sec^{-1} x\) is \([0, \pi] - \left\{\frac{\pi}{2}\right\}\). — Correct. \(\frac{\pi}{2}\) is left out because \(\sec\frac{\pi}{2}\) is not defined.
(D) The domain of \(\cot^{-1} x\) is \((0, \pi)\). — Incorrect. The domain is \(\mathrm{R}\); \((0, \pi)\) is its range.