If `z_1=x_1+iy,z_2=x_2+iy_2 and z_1 = (i(z_2+1))/(z_2-1)`, prove that `x_1^2+y_1^2-x_1= (x_2^2+y_2^2+2x_2-2y_2+1)/((x_2-1)^2+y_2^2)`

If x + y + z = xyz , prove that x(1 -y^(2)) (1- z^(2))+ y(1- z^(2))(1- x^(2)) +z(1-x^(2)) (1- y^(2)) = 4xyz .

If x+y+z=xzy , prove that : x(1-y^2) (1-z^2) + y(1-z^2) (1-x^2) + z (1-x^2) (1-y^2) = 4xyz .

If x+y+z=xzy , 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)

For x_(1), x_(2), y_(1), y_(2) in R if 0 lt x_(1)lt x_(2)lt y_(1) = y_(2) and z_(1) = x_(1) + i y_(1), z_(2) = x_(2)+ iy_(2) and z_(3) = (z_(1) + z_(2))//2, then z_(1) , z_(2) , z_(3) satisfy :

If _1=|1 1 1x^2y^2z^2x y z|a n d_2=|1 1 1y z z xx y x y z|, without expanding prove that _1=_2

If z=x+iy such that the argument of (z-1)/(z+1) is always (pi)/(4). Prove that x^(2)+y^(2)-2y=1

If sin^-1 x+sin^-1 y+sin^-1 z= pi prove that: x^4+y^4+z^4+4x^2y^2z^2=2(x^2y^2+y^2z^2+z^2x^20

If xy + yz + zx = 1 , show that x/(1-x^(2)) +y/(1-y^(2)) + z/(1-z^(2))= (4xyz)/((1-x^(2))(1-y^(2)) (1-z^(2)))

If cos^(-1) x + cos^(-1) y + cos^(-1) z = pi" , prove that " x^(2) + y^(2)+ z^(2) + 2xyz = 1 .

A variable complex number z=x+iy is such that arg (z-1)/(z+1)= pi/2 . Show that x^2+y^2-1=0 .