Prove 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) .

Prove that |z_1+z_2|^2 = |z_1|^2 + |z_2|^2 if z_1/z_2 is purely imaginary.

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)

If |z_1|=1,|z_2|=1 then prove that |z_1+z_2|^2+|z_1-z_2|^2 =4.

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)

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 u= sqrt(z_1 z_2) , prove that |z_1|+|z_2|=|(z_1+z_2)/2+u|+|(z_1+z_2)/2-u| .

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^(+) .