The value of cos^-1(cos 4) + tan^-1(tan 4) , where 4 is in radians, is

The correct answer is Option (2) → \(\pi\)

Taking \(\pi \approx 3.14\), the angle 4 lies between \(\pi\) and \(\frac{3\pi}{2} \approx 4.71\). It is outside both principal value branches, so each term needs an angle inside the branch with the same cosine or tangent.

For \(\cos^{-1}\) (branch \([0, \pi]\)): cosine does not change when the angle \(\theta\) is replaced by \(2\pi - \theta\).

    \(\cos(2\pi - 4) = \cos 4\), and \(2\pi - 4 \approx 2.28\) lies in \([0, \pi]\)

    \(\cos^{-1}(\cos 4) = 2\pi - 4\)

For \(\tan^{-1}\) (branch \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\)): tangent repeats after \(\pi\).

    \(\tan(4 - \pi) = \tan 4\), and \(4 - \pi \approx 0.86\) lies in \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\)

    \(\tan^{-1}(\tan 4) = 4 - \pi\)

Adding:

    \((2\pi - 4) + (4 - \pi) = \pi\)

Why the other options are incorrect

• Option (1) 8 — Incorrect. This takes both terms as 4, but 4 is outside both \([0, \pi]\) and \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\).

• Option (3) \(3\pi - 8\) — Incorrect. This is \((2\pi - 4) + (\pi - 4)\): it takes \(\tan^{-1}(\tan 4)\) as \(\pi - 4\), but \(\tan(\pi - 4) = -\tan 4\).

• Option (4) \(8 - 2\pi\) — Incorrect. This is \((4 - \pi) + (4 - \pi)\): it takes \(\cos^{-1}(\cos 4)\) as \(4 - \pi\), but \(\cos(4 - \pi) = -\cos 4\).