Which one of the following is TRUE?

The correct answer is Option (4) → \( \tan^{-1}1\lt\tan 1 \)

Here 1 is in radians (about 57°), while each inverse function of 1 is a standard angle.

Option (1) \( \cot^{-1}1\lt\cot 1 \) — Incorrect. \( \cot^{-1}1=\frac{\pi}{4}\approx 0.79 \), while \( \cot 1\approx 0.64 \).

Option (2) \( \cos^{-1}1\gt\cos 1 \) — Incorrect. \( \cos^{-1}1=0 \), while \( \cos 1\approx 0.54 \).

Option (3) \( \sin^{-1}1\lt\sin 1 \) — Incorrect. \( \sin^{-1}1=\frac{\pi}{2}\approx 1.57 \), while \( \sin 1\approx 0.84 \).

Option (4) \( \tan^{-1}1\lt\tan 1 \) — Correct. \( \tan^{-1}1=\frac{\pi}{4}\approx 0.79 \), while \( \tan 1\approx 1.56 \).

Shortcut: Since \( 1\gt\frac{\pi}{4} \) and tan is increasing on \( \left(0,\frac{\pi}{2}\right) \),

    \( \displaystyle \tan 1\gt\tan\frac{\pi}{4}=1 \)

    \( \displaystyle 1\gt\frac{\pi}{4}=\tan^{-1}1 \)

so \( \tan 1\gt\tan^{-1}1 \).