Observe that, #sqrt(1-x^2) and sqrt(1-y^2)# are Meaningful, iff,
#|x| le 1, and, |y| le 1...............(star^1).#
This means that, there is no Harm if we let,
#x=sintheta, and, y=sinphi....................(star^2).#
#:. sqrt(1-x^2)+sqrt(1-y^2)=a(x-y)," becomes, "#
# costheta+cosphi=a(sintheta-sinphi).#
#:.2cos((theta+phi)/2)cos((theta-phi)/2)=2a{cos((theta+phi)/2)sin((theta-phi)/2)}.#
#:.cos((theta-phi)/2)=asin((theta-phi)/2).#
# cot((theta-phi)/2)=a.#
# ((theta-phi)/2)=arc cota.#
#:. theta-phi=2arc cota.#
From, #(star^1), and, (star^2),# we have,
# arc sinx-arc siny=2arc cota.#
#:. d/dx{arc sinx-arc siny}=d/dx{2arc cota}.#
#:. 1/sqrt(1-x^2)-1/sqrt(1-y^2)*d/dx{y}=0,..."[The Chain Rule]."#
#:. 1/sqrt(1-x^2)-1/sqrt(1-y^2)*dy/dx=0.#
# :. dy/dx=sqrt(1-y^2)/sqrt(1-x^2).#
# rArr dy/dx=sqrt{(1-y^2)/(1-x^2)}.#
Enjoy Maths.!
