unit circle derived distribution

X_pdf(x) = 0.5
Y = f(X) = sqrt(1-X^2)

Y_pdf(y) =
Y_cdf'(y) =
( P( Y < y                               ) )' =
( P( f(X) < y                            ) )' =
( P( sqrt(1-X^2) < y                     ) )' =
( P( 1-X^2 < y^2                         ) )' =
( P( -1+X^2 > -y^2                       ) )' =
( P( X^2 > 1-y^2                         ) )' =
( P( sqrt(X^2) > sqrt(1-y^2)             ) )' =
( P( |X| > sqrt(1-y^2)                   ) )' =
( P( -X > sqrt(1-y^2) OR X > sqrt(1-y^2) ) )' =
( P( X < -sqrt(1-y^2) OR X > sqrt(1-y^2) ) )' =

by the OR rule, P(A OR B) = P(A) + P(B) - P(A AND B):

( P( X < -sqrt(1-y^2) ) + P( X > sqrt(1-y^2) ) - P( X < -sqrt(1-y^2) AND X > sqrt(1-y^2) ) )' =

by the AND rule, P(A AND B) = P(A) * P(B|A) = P(B) * P(A|B):

( P( X < -sqrt(1-y^2) ) + P( X > sqrt(1-y^2) ) - P( X < -sqrt(1-y^2) ) * P( X > sqrt(1-y^2) | X < -sqrt(1-y^2) ) )' =

the probability that X > sqrt(1-y^2) is zero, given that X < -sqrt(1-y^2)

( P( X < -sqrt(1-y^2) ) + P( X > sqrt(1-y^2) ) - P( X < -sqrt(1-y^2) ) * 0 )' =
( P( X < -sqrt(1-y^2) ) + P( X > sqrt(1-y^2) ) - 0 )' =
( P( X < -sqrt(1-y^2) ) + P( X > sqrt(1-y^2) ) )' =
( X_cdf( -sqrt(1-y^2) ) + (1-X_cdf( sqrt(1-y^2) )) )' =

performing the outer derivative:

X_cdf'( -sqrt(1-y^2) ) - X_cdf'( sqrt(1-y^2) ) =

performing the remaining derivatives using the chain rule:

X_pdf( -sqrt(1-y^2) ) * (-0.5*(1-y^2)^-0.5)*(-2*y) - X_pdf( sqrt(1-y^2) ) * (0.5*(1-y^2)^-0.5)*(-2*y) =
X_pdf( -sqrt(1-y^2) ) * y/sqrt(1-y^2) + X_pdf( sqrt(1-y^2) ) * y/sqrt(1-y^2) =
y/sqrt(1-y^2) * ( X_pdf(-sqrt(1-y^2)) + X_pdf(sqrt(1-y^2)) ) =
y/sqrt(1-y^2) * ( 0.5 + 0.5 ) =
y/sqrt(1-y^2)