The value of \(\sec^{-1}\left(\sec\displaystyle\frac{5\pi}{3}\right)\) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle\frac{\pi}{3}\)
The angle \(\frac{5\pi}{3}\) is not in the principal value branch \([0, \pi] - \left\{\frac{\pi}{2}\right\}\), so the answer is not \(\frac{5\pi}{3}\) itself.
The inner value is
\( \displaystyle \cos\frac{5\pi}{3} = \cos\left(2\pi - \frac{\pi}{3}\right) = \cos\frac{\pi}{3} \)
\( \displaystyle = \frac{1}{2} \)
So \(\sec\frac{5\pi}{3} = 2\). The angle in \([0, \pi] - \left\{\frac{\pi}{2}\right\}\) whose secant is 2 is \(\frac{\pi}{3}\), since \(\cos\frac{\pi}{3} = \frac{1}{2}\):
\( \displaystyle \sec^{-1}(2) = \frac{\pi}{3} \)
Why the other options are incorrect
Option (2) \(\frac{5\pi}{3}\) — Incorrect. \(\frac{5\pi}{3}\) is greater than \(\pi\), so it is outside the principal value branch.
Option (3) \(-\frac{\pi}{3}\) — Incorrect. \(\sec\left(-\frac{\pi}{3}\right) = 2\), but negative angles are outside \([0, \pi]\).
Option (4) \(\frac{2\pi}{3}\) — Incorrect. \(\cos\frac{2\pi}{3} = -\frac{1}{2}\), so \(\sec\frac{2\pi}{3} = -2\), not 2.