Why is the derivative of inverse tan(x) 1/(1+x^2)?

This can be proven by understanding tan(x) and it's inverse as functions, using implicit differentiation, subsitution and by recognising trigonometric identities (or being able to prove them from first principles). Once this is learned, the same can be repeated for the inverse of other trigonometric functions.  I have drafted a proof which can be seen in the Online Lesson Space.