Here,
#f(x)=sin(tan^-1(x)-sin^-1(x))#
Let,
#alpha=tan^-1x=>x=tanalpha,where,x > 0 ,alphain (-pi/2,pi/2)#
#beta=sin^-1x=>x=sinbeta,where,0 < x <1,betain[-pi/2,pi/2]#
So,
#f(x)=color(red)(sin(alpha-beta)=sinalphacosbeta-cosalphasinbeta...to(1)#
Now,
#sinalpha=sin(tan^-1x)=sin(sin^-1(x/sqrt(1+x^2)))=x/sqrt(1+x^
2#
#cosbeta=cos(sin^-1x)=cos(cos^-1(sqrt(1-x^2)))=sqrt(1-x^2#
#cosalpha=cos(tan^-1x)=cos(cos^-1(1/sqrt(1+x^2))=1/sqrt(1+x^2
)#
#sinbeta=sin(sin^-1x)=x#
From #color(red)((1)#
#f(x)=x/sqrt(1+x^2)xxsqrt(1-x^2)-1/sqrt(1+x^2)*x#
#=>f(x)=x/sqrt(1+x^2)[sqrt(1-x^2)-1]#
