`(y^(2)+yz+z^(2))/((x-y)(x-z))`+`(z^(2)+zx+x^(2))/((y-z)(y-x))`+`(x^(2)+xy+y^(2))/((z-x)(z-y))`

Simplify: ((x^(2)-y^(2))^(3) + (y^(2) - z^(2))^(3) + (z^(2) -x^(2))^(3))/((x-y)^(3) + (y-z)^(3) + (z-x)^(3))

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 x + 2y + 3z = 0 and x^(3) + 4y^(3) + 9z^(3)= 18 xyz , evaluate: ((x+2y)^(2))/(xy) + ((2y + 3z)^(2))/(yz) + ((3z+ x)^(2))/(zx)

If x+y+z=xyz , prove by trignometry that: (2x)/(1-x^(2))+(2y)/(1-y^(2))+(2z)/(1-z^(2))=(2x)/(1-x^(2)).(2y)/(1-y^(2)).(2z)/(1-z^(2))

Prove that : |{:((y+z)^(2),x^(2),x^(2)),(y^(2),(x+z)^(2),y^(2)),(z^(2),z^(2),(x+y)^(2)):}|=2xyz (x+y+z)^(3)

Show that: ((a^(x+1))/(a^(y+1)))^(x+y)\ ((a^(y+2))/(a^(z+2)))^(y+z)\ ((a^(z+3))/(a^(x+3)))^(z+x)=1

prove that: |(y^(2)z^(2),yz,y+z),(z^(2)x^(2),zx,z+x),(x^(2)y^(2),xy,x+y)|=0

If x^(3)+y^(3)+z^(3)=3xyz and x+y+z=0, find the value of ((x+y)^(2))/(xy)=((y+z)^(2))/(yz) +((z+x)^(2))/(zx)

Verify that x^(3)+y^(3)+z^(3)-3xyz=(1)/(2)(x+y+z)[(x-y)^(2)+(y-z)^(2)+(z-x)^(2)]

If x+y+z=x y z prove that (2x)/(1-x^2)+(2y)/(1-y^2)+(2z)/(1-z^2)=(2x)/(1-x^2)(2y)/(1-y^2)(2z)/(1-z^2)dot