Which of the following expressions is defined?

Correct answer: \(\frac{\cos^{-1}(-1)}{\cos^{-1}(-1)}\)

The domain of arccos, \(\cos^{-1}\), is \([-1,1]\). Thus \(\cos^{-1}(-1)\) is defined and equals \(\pi\), so the expression equals \(\pi/\pi=1\). \(\cos^{-1}(3)\) and \(\sin^{-1}(5/4)\) are undefined because 3 and 5/4 lie outside the domain \([-1,1]\). Arcsec has domain \(|x|\ge 1\), so \(\sec^{-1}(1/2)\) is also undefined. Exam tip: always check the allowed domain of the inverse trig function before simplifying; values outside the domain make the expression invalid regardless of algebraic form.