`z=x+iy`, prove that `((z)/bar(z)+(bar(z))/(z))=2((x^(2)-y^(2))/(x^(2)+y^(2)))`

For all z in C, prove that (i) (1)/(2)(z+bar(z))=Re(z), (ii) (1)/(2i)(z-bar(z))=Im(z), (iii) z bar(z)=|z|^(2), (iv) z+bar(z))"is real", (v) (z-bar(z))"is 0 or imaginary".

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

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

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)

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)

Prove that: R_e z= (z+bar(z))/2, I_m z= (z-bar(z))/(2i) .

If z = x + iy, then z bar(z) =

Prove that |(x^(2),x^(2)-(y-z)^(2),yz),(y^(2),y^(2)-(z-x)^(2),zx),(z^(2),z^(2)-(x-y)^(2),xy)|=(x-y)(y-z)(z-x)(x+y+z)(x^(2) + y^(2) + z^(2)) .

Prove that |{:(x^(2),,x^(2)-(y-z)^(2),,yz),(y^(2),,y^(2)-(z-x)^(2),,zx),(z^(2),,z^(2)-(x-y)^(2),,xy):}| =(x-y) (y-z) (z-x)(x+y+z) (x^(2)+y^(2)+z^(2))

Prove that |{:(x^(2),,x^(2)-(y-z)^(2),,yz),(y^(2),,y^(2)-(z-x)^(2),,zx),(z^(2),,z^(2)-(x-y)^(2),,xy):}| =(x-y) (y-z) (z-x)(x+y+z) (x^(2)+y^(2)+z^(2))