Step-by-Step Solution
Key Concept: Inverse trigonometric functions are multivalued; their principal branches satisfy $f(f^{-1}x) = x$ but $f^{-1}(f x) = x$ only when $x$ is in the appropriate range.
The inverse trigonometric functions have specific properties: $|\tan^{-1} x| = \tan^{-1}|x|$ for all $x \in \mathbb{R}$, $\sin(\sin^{-1}x) = \sin^{-1}|x| = |\sin^{-1}x|$ for all $|x| \leq 1$, and $\cot(\cot^{-1}x) = \cot^{-1}x = x$ by definition. However, $\tan^{-1}(\tan x) \neq x$ in general since $\tan$ is periodic. Therefore statements (A), (B), and (D) are correct, but (C) is incorrect.
Correct Answer: 1,2,4