cos function of random variable

Can any one please check whether below derivation is correct or not

Let X is uniformly distributed random variable as X ~U ([-π/2,π/2]). The probability density function $$f(x)= 1/\pi , -\pi/2\leq X \leq \pi/2$$ The distribution function of X is $$F_X(x)={2x+\pi\over 2\pi}, -\pi/2\lt X \lt \pi/2$$ We can further assume y=g(x)=cos(x) is a random variable (Y) where the function of a random variable is itself a random variable where ( y∈[0,1]). The PDF (g(y)) of y can be obtained by using change of variable (Jacobian) technique $$g(y)=f(x(y))|{dx\over dy}|$$ then get $$g(y)={1\over (\sqrt {1-y^2})}$$ The distribution function of Y can be given as: $$F_Y(y)=P(-1\lt X \lt1) ={2\over \pi}[ \sin^{-1} y]$$ after substitution y=0 and y=1 we get $$P(-1\lt X \lt1) =1$$

Regards