Prove that `z_(1)^(2)-z_(2)^(2)=(z_(1)+z_(2))(z_(1)-z_(2))`

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) and z_(2) are two complex numbers then prove that |z_(1)-z_(2)|^(2)<=(1+k)z_(1)^(2)+(1+(1)/(k))z_(2)^(2)

If z_(1) and z_(2) are two complex numbers,then (A) 2(|z|^(2)+|z_(2)|^(2)) = |z_(1)+z_(2)|^(2)+|z_(1)-z_(2)|^(2) (B) |z_(1)+sqrt(z_(1)^(2)-z_(2)^(2))|+|z_(1)-sqrt(z_(1)^(2)-z_(2)^(2))| = |z_(1)+z_(2)|+|z_(1)-z_(2)| (C) |(z_(1)+z_(2))/(2)+sqrt(z_(1)z_(2))|+|(z_(1)+z_(2))/(2)-sqrt(z_(1)z_(2))|=|z_(1)|+|z_(2)| (D) |z_(1)+z_(2)|^(2)-|z_(1)-z_(2)|^(2) = 2(z_(1)bar(z)_(2)+bar(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)|

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)|=......=|z_(n)|=1, prove that |z_(1)+z_(2)+z_(3)++z_(n)|=(1)/(z_(1))+(1)/(z_(2))+(1)/(z_(3))++(1)/(z_(n))

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 z_(1),z_(2)inC , show that (z_(1)+z_(2))^(2)=z_(1)^(2)+2z_(1)z_(2)+z_(2)^(2) .

If z_(1) and z_(2) are two complex numbers and c>0 then prove that (z_(1)+z_(2))^(2)<=(1+c)|z_(1)|^(2)+(1+c^(-1))|z_(2)|^(2)