`z_1 8z z_2,|1-z_1( & z )_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 .

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)

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))

Which of the following are correct for any two complex numbers z_1 and z_2? (A) |z_1z_2|=|z_1||z_2| (B) arg(|z_1 z_2|)=(argz_1)(arg,z_2) (C) |z_1+z_2|=|z_1|+|z_2| (D) |z_1-z_2|ge|z_1|-|z_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|=1,|z_2|=1 then prove that |z_1+z_2|^2+|z_1-z_2|^2 =4.

z_ (1) 8zz_ (2), | 1-z_ (1) (& z) _ (2) | ^ (2) - | z_ (1) -z_ (2) | ^ (2) = (1- | z_ (1) | ^ (2)) (1- | z_ (2) | ^ (2))