Here,
#color(red)(y=cos(sinx).......................to(I)#
Diff.w.r.t.x , using Chain Rule
#(dy)/(dx)=-sin(sinx)d/(dx)(sinx)#
#color(red)((dy)/(dx)=-sin(sinx)cosx....to(II)#
Again diff.w.r.t.x, using Product and Chain Rule,
#(d^2y)/(dx^2)=-[sin(sinx)d/(dx)(cosx)+cosxd/(dx)(sin(sinx)) ]#
#(d^2y)/(dx^2)=-[sin(sinx)(-sinx)+cosxcos(sinx)cosx]#
#(d^2y)/(dx^2)=sin(sinx)(sinx)-cos(sinx)cos^2x]#
#(d^2y)/(dx^2)=sin(sinx)color(blue)cosx(sinx/color(blue)cosx)-
cos(sinx)cos^2x#
Using #(I) and(II),#we get
#(d^2y)/(dx^2)=(-(dy)/(dx))(tanx)-ycos^2x#
#(d^2y)/(dx^2)+tanx(dy)/(dx)+ycos^2x=0#