Prove that `|1-z_(1)z_(2)|^(2)-|z_(1)-z_(2)|^(2)=(1-|z_(1)|^(2))(1-|z_(2)|^(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 any two complex numbers show that |z_(1)+z_(2)|^(2)+|z_(1)-z_(2)|^(2)=2|z_(1)|^(2)+2|z_(2)|^(2)

Prove that |1-barz_1z_2|^2-|z_1-z_2|^2=(1-|z_1|^2)(1-|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)

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))

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) 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)

Prove the identity, | 1-z_ (1) bar (z) _ (2) | ^ (2) - | z_ (1) -z_ (2) | ^ (2) = (1- | z_ (1) | ^ (2)) (1- | z_ (2) | ^ (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)|