Question

cos[arcsin(2√x/1+x)]

Answer 1

we have that x>0 and 2sqrt(x)<=1+x for every x>0
cos(t)=sqrt(1-sin^2(t))
cos [arcsin(2vx/1+x)]=sqrt( 1-sin^2[arcsin(2vx/1+x)])=sqrt( 1- ( sin[arcsin(2vx/1+x)])^2 )=sqrt( 1- (2vx/(1+x))^2)=
=sqrt(1-4x/(1+x)^2)