Prove that `|(z_(1))/(|z_(1)|)+(z_(2))/(|z_(2)|)|<=2`

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

If z_(1),z_(2) and z_(3),z_(4) are two pairs of conjugate complex numbers,prove that arg ((z_(1))/(z_(4)))+arg((z_(2))/(z_(3)))=0

If z_(1) and z_(2) are two complex number such that |z_(1)|<1<|z_(2)| then prove that |(1-z_(1)bar(z)_(2))/(z_(1)-z_(2))|<1

If z_(1) and z_(2) two different complex numbers and |z_(2)|=1 then prove that |(z_(2)-z_(1))/(1-bar(z)_(1)z_(2))|=1

For all complex numbers z_(1) and z_(2) , prove that |z_(1)+z_(2)|^(2)+|z_(1)-z_(2)|^(2)=2(|z_(1)|^(2)+|z_(2)|^(2)) .

if |z_(1)+z_(2)|=|z_(1)|+|z_(2)|, then prove that arg(z_(1))=arg(z_(2)) if |z_(1)-z_(2)|=|z_(1)|+|z_(2)| then prove that arg (z_(1))=arg(z_(2))=pi

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)|

If z_(1) and z_(2) are two complex numbers such that |z_(1)| lt 1 lt |z_(2)| , then prove that |(1- z_(1)barz_(2))//(z_(1)-z_(2))| lt 1

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 .

If complex numbers z_(1)z_(2) and z_(3) are such that |z_(1)| = |z_(2)| = |z_(3)| , then prove that arg((z_(2))/(z_(1)) = arg ((z_(2) - z_(3))/(z_(1) - z_(3)))^(2)