(*********************************************************)
(** * Definition of arcsine based on arctangent          *)
(*********************************************************)
(** asin is defined by cases so that it is defined in the full range from -1 .. 1 *)
Definition asin x :=
if Rle_dec x (-1) then - (PI / 2) else
if Rle_dec 1 x then PI / 2 else
atan (x / sqrt (1 - x²)).
(** ** Relation between arcsin and arctangent *)

Lemma asin_atan : forall x, -1 < x < 1 ->
asin x = atan (x / sqrt (1 - x²)).