The value of \(\mathrm{cosec}^{-1}\left(\mathrm{cosec}\,\displaystyle\frac{3\pi}{4}\right)\) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{\pi}{4}\)
The principal value branch of \(\mathrm{cosec}^{-1}\) is \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] - \{0\}\). The angle \(\frac{3\pi}{4}\) is outside it, so first we find the value of \(\mathrm{cosec}\,\frac{3\pi}{4}\).
Sine (and so cosecant) does not change when the angle is replaced by \(\pi\) minus the angle:
\( \displaystyle \mathrm{cosec}\,\frac{3\pi}{4} = \mathrm{cosec}\left(\pi - \frac{3\pi}{4}\right) \)
\( \displaystyle = \mathrm{cosec}\,\frac{\pi}{4} = \sqrt{2} \)
Since \(\frac{\pi}{4}\) lies in the branch,
\( \displaystyle \mathrm{cosec}^{-1}\left(\mathrm{cosec}\,\frac{3\pi}{4}\right) = \mathrm{cosec}^{-1}\left(\sqrt{2}\right) = \frac{\pi}{4} \)
Why the other options are incorrect
Option (1) \(\frac{3\pi}{4}\) — Incorrect. \(\frac{3\pi}{4}\) is greater than \(\frac{\pi}{2}\), so it cannot be a value of \(\mathrm{cosec}^{-1}\).
Option (3) \(-\frac{\pi}{4}\) — Incorrect. \(\mathrm{cosec}\left(-\frac{\pi}{4}\right) = -\sqrt{2}\), but \(\mathrm{cosec}\,\frac{3\pi}{4} = +\sqrt{2}\).
Option (4) \(\frac{5\pi}{4}\) — Incorrect. \(\frac{5\pi}{4}\) is outside the branch, and \(\mathrm{cosec}\,\frac{5\pi}{4} = -\sqrt{2}\).