The value of \(\tan^{-1}\left(\tan\displaystyle\frac{4\pi}{5}\right)\) is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(-\displaystyle\frac{\pi}{5}\)
\(\tan^{-1}(\tan x) = x\) only when x lies in the principal value branch \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\). Here \(\frac{4\pi}{5}\) is outside it.
Tangent has period \(\pi\), so
\( \displaystyle \tan\frac{4\pi}{5} = \tan\left(\frac{4\pi}{5} - \pi\right) \)
\( \displaystyle = \tan\left(-\frac{\pi}{5}\right) \)
Since \(-\frac{\pi}{5}\) lies in \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\),
\( \displaystyle \tan^{-1}\left(\tan\frac{4\pi}{5}\right) = -\frac{\pi}{5} \)
Why the other options are incorrect
Option (1) \(\frac{4\pi}{5}\) — Incorrect. \(\frac{4\pi}{5}\) is greater than \(\frac{\pi}{2}\), so it is not a value of \(\tan^{-1}\).
Option (2) \(\frac{\pi}{5}\) — Incorrect. \(\tan\frac{\pi}{5}\) is positive, but \(\tan\frac{4\pi}{5}\) is negative (second quadrant).
Option (3) \(-\frac{4\pi}{5}\) — Incorrect. \(-\frac{4\pi}{5}\) is less than \(-\frac{\pi}{2}\), so it is outside the principal branch.