What is the range of the function (\tan x)?

Correct answer: \(( -\infty,\infty )\)

\(\tan x=\dfrac{\sin x}{\cos x}\) is defined wherever \(\cos x\ne0\). On every interval where it is defined, \(\tan x\) takes every real value from \(-\infty\) to \(\infty\). Hence its range is \(( -\infty,\infty )\). The interval \([-1,1]\) is the range of \(\sin x\) and \(\cos x\), not of \(\tan x\). Exam tip: the ranges of \(\tan x\) and \(\cot x\) are all real numbers.