Prove the identity,`|1-z_1barz_2|^2-|z_1-z_2|^2=(1-|z_1|^2)(1-|z_2|^2)`

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

Prove that |z_1+z_2|^2+|z_1-z_2|^2 =2|z_1|^2+2|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)) .

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

Let z_1=r_1(costheta_1+isintheta_1)a n dz_2=r_2(costheta_2+isintheta_2) be two complex numbers. Then prove that |z_1+z_2|^2=r1 2+r2 2+2r_1r_2cos(theta_1-theta_2) or |z_1+z_2|^2=|z_1|^2+|z_2|^2+2|z_1||z_2|^()_cos(theta_1-theta_2) |z_1-z_2|^2=r1 2+r2 2-2r_1r_2cos(theta_1-theta_2) or |z_1-z_2|^2=|z_1|^2+|z_2|^2-2|z_1||z_2|^()_cos(theta_1-theta_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 prove that |z_1 - z_2|^2 le (1+ k)|z_1|^2 +(1+K^(-1) ) |z_2|^2 AA k in R^(+) .

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