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(** * Definition of arcsine based on arctangent          *)
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(** 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)).

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