Match List-I with List-II

Choose the correct answer from the options given below:

(A)-(II), (B)-(III), (C)-(IV), (D)-(I)

The correct answer is Option (3) → (A)-(II), (B)-(III), (C)-(IV), (D)-(I)

(A) $\sin^{-1}\left(-\frac{1}{2}\right)$ Principal value lies in $[-\frac{\pi}{2},\frac{\pi}{2}]$. $\sin^{-1}\left(-\frac{1}{2}\right) = -\frac{\pi}{6}$ → (A) → (II)

(B) $\cos^{-1}\left(-\frac{1}{2}\right)$ Principal value lies in $[0,\pi]$. $\cos^{-1}\left(-\frac{1}{2}\right) = \frac{2\pi}{3}$ → (B) → (III)

(C) $\tan^{-1}(-\sqrt{3})$ Principal value lies in $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$. $\tan^{-1}(\sqrt{3})=\frac{\pi}{3}$ → negative gives $\tan^{-1}(-\sqrt{3})=-\frac{\pi}{3}$ → (C) → (IV)

(D) $\cot^{-1}(\sqrt{3})$ Principal value lies in $(0,\pi)$. $\cot^{-1}(\sqrt{3})=\frac{\pi}{6}$ → (D) → (I)

Final matching:

(A)–(II), (B)–(III), (C)–(IV), (D)–(I)