What is the value of \(\sec^{-1}\left(-1\right)\)?

Correct answer: \(0\)

Since \(\sec\theta=\dfrac{1}{\cos\theta}\) and \(\sec 0=1\), the principal value of \(\sec^{-1}1\) is \(0\). The common principal range for \(\sec^{-1}x\) is \([0,\pi]\) excluding \(\pi/2\), so \(\sec^{-1}1=0\). Option \(\pi\) is incorrect because \(\sec\pi=-1\). Options \(\tfrac{\pi}{2}\) and \(-\tfrac{\pi}{2}\) are invalid since \(\sec\) is undefined where \(\cos\theta=0\). Exam tip: always check the principal range and where the original trig function is defined before selecting the inverse value.