Match List-I with List-II
| List-I | List-II | ||
|---|---|---|---|
| Expression | Principal value | ||
| (A) | \(\sec^{-1}(-2)\) | (I) | \(\pi\) |
| (B) | \(\mathrm{cosec}^{-1}\left(\sqrt{2}\right)\) | (II) | \(-\displaystyle\frac{\pi}{4}\) |
| (C) | \(\tan^{-1}(-1)\) | (III) | \(\displaystyle\frac{2\pi}{3}\) |
| (D) | \(\cos^{-1}(-1)\) | (IV) | \(\displaystyle\frac{\pi}{4}\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A)-(III), (B)-(IV), (C)-(II), (D)-(I)
Each value must lie in the principal value branch: \([0, \pi] - \left\{\frac{\pi}{2}\right\}\) for \(\sec^{-1}\), \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] - \{0\}\) for \(\mathrm{cosec}^{-1}\), \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\) for \(\tan^{-1}\) and \([0, \pi]\) for \(\cos^{-1}\).
(A) \(\sec^{-1}(-2)\)
This matches (III) \(\frac{2\pi}{3}\), the second-quadrant angle in \([0, \pi]\):
\( \displaystyle \sec\frac{2\pi}{3} = \frac{1}{\cos\frac{2\pi}{3}} = \frac{1}{-\frac{1}{2}} = -2 \)
(B) \(\mathrm{cosec}^{-1}\left(\sqrt{2}\right)\)
This matches (IV) \(\frac{\pi}{4}\):
\( \displaystyle \mathrm{cosec}\,\frac{\pi}{4} = \frac{1}{\sin\frac{\pi}{4}} = \sqrt{2} \)
(C) \(\tan^{-1}(-1)\)
This matches (II) \(-\frac{\pi}{4}\), a negative angle because the number is negative:
\( \displaystyle \tan\left(-\frac{\pi}{4}\right) = -1 \)
(D) \(\cos^{-1}(-1)\)
This matches (I) \(\pi\), the right end of \([0, \pi]\):
\( \displaystyle \cos\pi = -1 \)