Which statement is correct: what does (\sin^{-1}x) mean?

Correct answer: It is the inverse function of (\sin x)

The direct answer is B: \(\sin^{-1}x\) is the inverse function of sine, with the usual restricted sine domain used to make an inverse possible. Inverse means reversing an operation. If \(y=\sin^{-1}x\), then \(\sin y=x\), and the principal value of y is chosen in \([-\frac{\pi}{2},\frac{\pi}{2}]\). This is different from taking a reciprocal. Option A, \(\frac{1}{\sin x}\), is the reciprocal of \(\sin x\), called \(\cosec x\), not inverse sine. Option B is correct because it describes the inverse function: it takes a sine value as input and returns the corresponding principal angle. Option C is wrong because \(\cosec x=\frac{1}{\sin x}\), whereas inverse sine returns an angle; these are different kinds of quantities and are not always equal. Option D is wrong because inverse sine is defined for every real x in \([-1,1]\), not only for x=0. For example, \(\sin^{-1}(1)=\frac{\pi}{2}\). Memory cue: superscript -1 on a function may mean inverse, while reciprocal is written as \(1/\sin x\); read the notation carefully.