The value of \(\tan\left(\cos^{-1}\left(-\displaystyle\frac{1}{2}\right)\right)\) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(-\sqrt{3}\)
The principal value branch of \(\cos^{-1}\) is \([0, \pi]\). For a negative number the angle is in the second quadrant:
\( \displaystyle \cos^{-1}\left(-\frac{1}{2}\right) = \pi - \frac{\pi}{3} = \frac{2\pi}{3} \)
Tangent is negative in the second quadrant:
\( \displaystyle \tan\frac{2\pi}{3} = -\tan\frac{\pi}{3} = -\sqrt{3} \)
Why the other options are incorrect
Option (2) \(\sqrt{3}\) — Incorrect. This is \(\tan\frac{\pi}{3}\); it takes \(\cos^{-1}\left(-\frac{1}{2}\right)\) as \(\frac{\pi}{3}\), but \(\cos\frac{\pi}{3} = +\frac{1}{2}\).
Option (3) \(-\frac{1}{\sqrt{3}}\) — Incorrect. This is \(\cot\frac{2\pi}{3}\), not \(\tan\frac{2\pi}{3}\).
Option (4) \(\frac{1}{\sqrt{3}}\) — Incorrect. This is \(\cot\frac{\pi}{3}\); both the angle and the ratio are wrong.