Inverse Trigonometric Functions

We know that domain of the cot function (cotangent function) is the set \(\{x : x \in R \text{ and } x \neq n\pi, n \in Z\}\) and range is R. It means that cotangent function is not defined for integral multiples of \(\pi\). If we restrict the domain of cotangent function to \((0, \pi)\), then it is bijective with its range as R. In fact, cotangent function restricted to any of the intervals \((-\pi, 0), (0, \pi), (\pi, 2\pi)\) etc., is bijective and its range is R. Thus \(cot^{-1}\) can be defined as a function whose domain is the R and range as any of the intervals \((-\pi, 0), (0, \pi), (\pi, 2\pi)\) etc. These intervals give different branches of the function \(cot^{-1}\). The function with range \((0, \pi)\) is called the principal value branch of the function \(cot^{-1}\). We thus have \(cot^{-1}: R \rightarrow (0, \pi)\)