The value of \(\sec^{-1}(2) + \mathrm{cosec}^{-1}\left(-\sqrt{2}\right)\) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle\frac{\pi}{12}\)
The principal value branch of \(\sec^{-1}\) is \([0, \pi] - \left\{\frac{\pi}{2}\right\}\). Since \(\cos\frac{\pi}{3} = \frac{1}{2}\),
\( \displaystyle \sec^{-1}(2) = \frac{\pi}{3} \)
The principal value branch of \(\mathrm{cosec}^{-1}\) is \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] - \{0\}\), so a negative cosecant needs a negative angle. Since \(\sin\left(-\frac{\pi}{4}\right) = -\frac{1}{\sqrt{2}}\),
\( \displaystyle \mathrm{cosec}^{-1}\left(-\sqrt{2}\right) = -\frac{\pi}{4} \)
Adding:
\( \displaystyle \frac{\pi}{3} - \frac{\pi}{4} = \frac{4\pi - 3\pi}{12} \)
\( \displaystyle = \frac{\pi}{12} \)
Why the other options are incorrect
Option (2) \(\frac{7\pi}{12}\) — Incorrect. This is \(\frac{\pi}{3} + \frac{\pi}{4}\): it takes \(\mathrm{cosec}^{-1}\left(-\sqrt{2}\right)\) as \(\frac{\pi}{4}\), but \(\mathrm{cosec}\frac{\pi}{4} = +\sqrt{2}\).
Option (3) \(-\frac{\pi}{12}\) — Incorrect. This is \(\frac{\pi}{6} - \frac{\pi}{4}\): it takes \(\sec^{-1}(2)\) as \(\frac{\pi}{6}\), but \(\sec\frac{\pi}{6} = \frac{2}{\sqrt{3}}\).
Option (4) \(\frac{19\pi}{12}\) — Incorrect. This is \(\frac{\pi}{3} + \frac{5\pi}{4}\): \(\mathrm{cosec}\frac{5\pi}{4} = -\sqrt{2}\), but \(\frac{5\pi}{4}\) is outside \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\).