Prove that `|(z_1)/(z_2)|=|z_1|/|z_2|`

For any two complex numbers z_(1) and z_(2), prove that |z_(1)+z_(2)| =|z_(1)|-|z_(2)| and |z_(1)-z_(2)|>=|z_(1)|-|z_(2)|

For any two complex number z_(1) and z_(2) prove that: |z_(1)+z_(2)|>=|z_(1)|-|z_(2)|

For any two complex number z_(1) and z_(2) prove that: |z_(1)-z_(2)|>=|z_(1)|-|z_(2)|

For any two complex number z_(1) and z_(2) prove that: |z_(1)+z_(2)|<=|z_(1)|+|z_(2)|

For any two complex number z_(1) and z_(2) prove that: |z_(1)-z_(2)|<=|z_(1)|+|z_(2)|

If z_(1) and z_(2)(ne0) are two complex numbers, prove that: (i) |z_(1)z_(2)|=|z_(1)||z_(2)| (ii) |(z_(1))/(z_(2))|=(|z_(1)|)/(|z_(2)|),z_(2)ne0 .

Prove that |z_1+z_2|^2 = |z_1|^2 + |z_2|^2 if z_1/z_2 is purely imaginary.

If z_(1) and z-2 are complex numbers and u=sqrt(z_(1)z_(2)), then prove that |z_(1)|+|z_(2)|=|(z_(1)+z_(2))/(2)+u|+|(z_(1)+z_(2))/(2)-u|

If u= sqrt(z_1 z_2) , prove that |z_1|+|z_2|=|(z_1+z_2)/2+u|+|(z_1+z_2)/2-u| .

Prove that |z_(1)+z_(2)|^(2)=|z_(1)|^(2)+|z_(2)|^(2),quad if z_(1)/z_(2) is purely imaginary.