} is called the principal value branch of the function sec – . We thus have sec – : R – (– , ) → [ , π ] – { π } The graphs of the functions y = sec x and y = sec - x are given in Fig . (i), (ii). Finally, we now discuss tan – and cot – We know that the domain of the tan function (tangent function) is the set { x : x ∈ R and x ≠ ( n + ) π , n ∈ Z } and the range is R .
It means that tan function is not defined for odd multiples of π . If we restrict the domain of tangent function to Fig . (i) Fig . (ii) , then it is one-one and onto with its range as R .
Actually, tangent function restricted to any of the intervals , −π −π etc., is bijective and its range is R . Thus tan – can be defined as a function whose domain is R and range could be any of the intervals , −π −π , , and so on. These intervals give different branches of the function tan – . The branch with range is called the principal value branch of the function tan – .
We thus have tan – : R → The graphs of the function y = tan x and y = tan – x are given in Fig . (i), (ii). Fig . (i) Fig .
(ii) We know that domain of the cot function (cotangent function) is the set { x : x ∈ R and x ≠ n π , n ∈ Z } and range is R . It means that cotangent function is not defined for integral multiples of π . If we restrict the domain of cotangent function to ( , π ), then it is bijective with and its range as R . In fact, cotangent function restricted to any of the intervals (– π , ), ( , π