Which is the correct form of \(\tan^{-1}(-x)\)?

Correct answer: (-\tan^{-1}x)

The inverse tangent function is an odd function. For an odd function, changing the sign of the input changes the sign of the output, so \(f(-x)=-f(x)\). Applying this property to the inverse tangent function gives \(\tan^{-1}(-x)=-\tan^{-1}(x)\). Therefore option A is correct.

The principal value of \(\tan^{-1}(x)\) lies in \((-\frac{\pi}{2},\frac{\pi}{2})\), and the negative of a principal value remains in the same interval. This confirms that no extra \(\pi\) or \(\frac{\pi}{2}\) term is needed. Option B would incorrectly ignore the negative input, while options C and D are not the odd-function identity and generally give a different value. The identity follows because tangent itself is odd and its inverse preserves the corresponding sign relationship on its principal domain.