If $z_1$ and $z_2$ are two complex numbers such that $|z_1|=1$ and $|z_2|=1$, then the value of $|z_1+z_2{|}^2+|z_1-z_2{|}^2$ is

Solution

1. Recall the identity $|z_1+z_2{|}^2=|z_1{|}^2+|z_2{|}^2+2\text{Re}(z_1{\bar{z}}_2)$.
2. Recall the identity $|z_1-z_2{|}^2=|z_1{|}^2+|z_2{|}^2-2\text{Re}(z_1{\bar{z}}_2)$.
3. Adding these two equations gives $|z_1+z_2{|}^2+|z_1-z_2{|}^2=2(|z_1{|}^2+|z_2{|}^2)$.
4. Since $|z_1|=1$ and $|z_2|=1$, the sum is $2(1 + 1) = 4$.
5. Hence the answer is (D).