For `|z-1|=1,` show that `tan{[a r g(z-1)]/2}-((2i)/z)=-i`

For any two complex number z_(1) and z_(2) , such that |z_(1)| = |z_(2)| = 1 and z_(1) z_(2) ne -1 , then show that (z_(1) + z_(2))/(1 + z_(1)z_(2)) is real number.

Find the center of the arc represented by a r g[(z-3i)//(z-2i+4)]=pi//4 .

If z_(1) = 3, z_(2) = -7i, and z_(3) = 5 + 4i, show that z_(1)(z_(2) + z_(3)) = z_(1) z_(2) + z_(1) z_(3)

If z_(1) = 1 - 3i, z_(2) = -4i and z_(3) = 5, show that (z_(1)z_(2)) z_(3) = z_(1) (z_(2)z_(3))

If z_(1) = 1 - 3i, z_(2) = -4i and z_(3) = 5, show that (z_(1) + z_(2)) + z_(3) = z_(1) + (z_(2) + z_(3))

If z_(1) = 3, z_(2) = -7i, and z_(3) = 5 + 4i, show that (z_(1) + z_(2)) z_(3) = z_(1) z_(3) + z_(2) z_(3)

If z_1a n dz_2 are two complex numbers such that |z_1|=|z_2|a n d arg(z_1)+a r g(z_2)=pi , then show that z_1,=-( barz )_2dot