The principal inverse tangent and cotangent functions for complex arguments can be defined as formulas involving principal natural logarithms, but these are not odd on the imaginary axis, which they must be according to their definitions as inverse functions. These formulas are therefore modified in such a way that they become odd on the imaginary axis, by choosing the other branch on the lower branch cut, and the corresponding addition formulas for complex and real arguments are derived. With these addition formulas their values on their branch cuts are determined, confirming these modified formulas. Some new formulas for the (hyperbolic) inverse tangent and cotangent functions for complex arguments and some new addition formulas for these functions for real arguments are derived. Some new formulas for the inverse sine and cosine functions and their connections with the inverse tangent and cotangent functions for complex arguments are provided, and from these some new addition formulas for the inverse sine and cosine functions for real arguments are derived. Some duplication and bisection formulas for the inverse tangent, cotangent, sine and cosine functions are derived.