What is (\tan^{-1}(-x)) equal to when (x\in\mathbb{R})?

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

The direct answer is A: \(\tan^{-1}(-x)=-\tan^{-1}x\). The symbol \(\tan^{-1}\) here means the inverse tangent function, not the reciprocal. A function is called odd when changing its input from \(x\) to \(-x\) changes its output to its negative. In general, if \(y=\tan^{-1}x\), then \(\tan y=x\), where the principal value of \(y\) lies between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\). Since tangent is an odd function, \(\tan(-y)=-\tan y=-x\). Therefore the principal inverse-tangent value for \(-x\) is \(-y=-\tan^{-1}x\). Option A is correct because it states this exact identity. Option B, \(\pi-\tan^{-1}x\), is not generally in the principal range and does not give the inverse tangent of a negative input. Option C uses a shift by \(\frac{\pi}{2}\), which has no valid basis here. Option D gives \(\tan^{-1}x\), ignoring the sign change, so it is wrong except in the special case \(x=0\). Memory cue: inverse tangent is odd, so simply put a minus sign outside.