Arctangent in terms of Arcsine/Proof 2

Proof
From Pfaff's Transformation:


 * $\ds \map F {a, b; c; x} = \paren {1 - x}^{-a} \map F {a, c - b; c; \dfrac x {x - 1} }$

where $\map F {a, b; c; x}$ is the Gaussian hypergeometric function of $x$.

We have:

Therefore:
 * $\map \arctan x = \map \arcsin {\dfrac x {\sqrt {1 + x^2} } }$