Consider the following statements:
(A) \(\cos^{-1} x \gt \displaystyle\frac{\pi}{2}\) for every x in \([-1, 0)\).
(B) \(\tan^{-1} x \lt 0\) for every \(x \lt 0\).
(C) \(\cot^{-1} x \lt \displaystyle\frac{\pi}{2}\) for every \(x \lt 0\).
(D) \(\sin^{-1} x \gt 0\) for every x in \([-1, 0)\).
Which of the above statements are correct?
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A) and (B) only
Each value can be compared with the value at \(x = 0\), since each function is either increasing or decreasing.
\( \displaystyle \cos^{-1} 0 = \frac{\pi}{2}, \quad \cot^{-1} 0 = \frac{\pi}{2} \) (both decreasing)
\( \displaystyle \tan^{-1} 0 = 0, \quad \sin^{-1} 0 = 0 \) (both increasing)
So for negative x, \(\cos^{-1} x\) and \(\cot^{-1} x\) are greater than \(\frac{\pi}{2}\), while \(\tan^{-1} x\) and \(\sin^{-1} x\) are negative.
Checking the statements
(A) \(\cos^{-1} x \gt \frac{\pi}{2}\) for every x in \([-1, 0)\). — Correct. \(\cos^{-1}\) is decreasing, so \(x \lt 0\) gives a value above \(\cos^{-1} 0 = \frac{\pi}{2}\); for example \(\cos^{-1}(-1) = \pi\).
(B) \(\tan^{-1} x \lt 0\) for every \(x \lt 0\). — Correct. \(\tan^{-1}\) is increasing, so \(x \lt 0\) gives a value below \(\tan^{-1} 0 = 0\); for example \(\tan^{-1}(-1) = -\frac{\pi}{4}\).
(C) \(\cot^{-1} x \lt \frac{\pi}{2}\) for every \(x \lt 0\). — Incorrect. \(\cot^{-1}(-1) = \frac{3\pi}{4}\), which is greater than \(\frac{\pi}{2}\).
(D) \(\sin^{-1} x \gt 0\) for every x in \([-1, 0)\). — Incorrect. \(\sin^{-1}\left(-\frac{1}{2}\right) = -\frac{\pi}{6}\), which is negative.