Trigonometry/Derivative of Inverse Functions
The inverse functions sin-1(x), etc. have derivatives that are purely algebraic functions.
If y = sin-1(x) then x = sin(y) and
If y = tan-1(x) then x = tan(y) and
If y = sec-1(x) then x = sec(y) and
The above results provide an easy way to find the power series expansions of these functions.
This is uniformly convergent if |x| < 1 so can be integrated term by term. The constant of integration is zero since sin-1(0) = 0, so
This is uniformly convergent if |x| < 1 so can be integrated term by term. The constant of integration is zero since tan-1(0) = 0, so
Note that sec-1(x) has no power series expansion about x=0, as it is not defined for x < 1 and has an infinite derivative when x = 1. An expansion about any point x = a > 1 in powers of (x-a) can be found uding Taylor's theorem; it will converge for 1 < x < 2a-1.