If `x^2=y^3`, then prove that `(x/y)^(3/2)+(y/x)^-(2/3)=x^(1/2)+y^(1/3)`.

If S_(n)=(x+y)+(x^(2)+xy+y^(2))+(x^(3)+x^(2)y+y^(2)x+y^(3))+…n terms then prove that (x-y)S_(n)=[(x^(2)(x^(n)-1))/(x-1)-(y^(2)y^(n)-1)/(y-1)] .

If x=1&y=-3 then prove the identity (x+y)^2=x^2+2xy+y^2

If x+y+z=xyz then prove that (3x-x^3)/(1-3x^2)+(3y-y^3)/(1-3y^2)+(3z-z^3)/(1-3z^2)=(3x-x^3)/(1-3x^2).(3y-y^3)/(1-3y^2).(3z-z^3)/(1-3z^2)

If (x^3+y^3) alpha(x^3-y^3) prove that (x^2+y^2) alpha(x^2-y^2) .

If x + y + z = xyz , prove that (3x -x^(3))/ (1-3x^(2)) + (3y -y^(3))/(1- 3y^(2)) +(3z -z^(3))/(1- 3z^(2)) = (3x -x^(3))/(1-3x)^(2) * (3y- y^(3))/(1-3x)^(2)* (3z- z^(3))/(1-3z)^(2) .

If the join of (x_(1),y_(1)) and (x_(2),y_(2)) makes on obtuse angle at (x_(3),y_(3)), then prove than (x_(3)-x_(1))(x_(3)-x_(2))+(y_(3)-y_(1))(y_(3)-y_(2))<0

If A(x_(1), y_(1)), B(x_(2), y_(2)) , and, C(x_(3), y_(3)) are vertices of an equilateral triangle whose each side is equal to a, then prove that |(x_(1), y_(1), 2),(x_(2), y_(2), 2),(x_(3), y_(3), 2)|^(2) = 3a^(4) .

If x+y+z=xyz , prove that: a) (3x-x^(3))/(1-3x^(2))+(3y-y^(3))/(1-3y^(2))+(3z-z^(3))/(1-3z^(2))= (3x-x^(3))/(1-3x^(2)).(3y-y^(3))/(1-3y^(2)).(3z-z^(3))/(1-3z^(2)) b) (x+y)/(1-xy) + (y+z)/(1-yz)+(z+x)/(1-zx)= (x+y)/(1-xy) .(y+z)/(1-yz).(z+x)/(1-zx)

If the tangent at (x_(1),y_(1)) to the curve x^(3)+y^(3)=a^(3) meets the curve again in (x_(2),y_(2)), then prove that (x_(2))/(x_(1))+(y2)/(y_(1))=-1