We are given two complex numbers $z_1$ and $z_2$ such that:

$|z_1 + z_2| = \sqrt{3}, \quad |z_1| = 1, \quad |z_2| = 1$

The problem asks us to determine the value of $|z_1 - z_2|$.

Step-by-step solution:

1. First, we can represent the complex numbers $z_1$ and $z_2$ in polar form. Since $|z_1| = 1$ and $|z_2| = 1$, we can write them as: $z_1 = e^{i\theta_1}, \quad z_2 = e^{i\theta_2}$ where $\theta_1$ and $\theta_2$ are the arguments of $z_1$ and $z_2$, respectively.

2. The modulus of the sum $z_1 + z_2$ is given as $|z_1 + z_2| = \sqrt{3}$. Using the polar form, we can write: $|z_1 + z_2| = |e^{i\theta_1} + e^{i\theta_2}| = \sqrt{3}$

3. Now, using the identity for the sum of two complex numbers in polar form: $|z_1 + z_2|^2 = |z_1|^2 + |z_2|^2 + 2 \cdot |z_1| \cdot |z_2| \cdot \cos(\theta_1 - \theta_2)$ Since $|z_1| = 1$ and $|z_2| = 1$, this simplifies to: $|z_1 + z_2|^2 = 1 + 1 + 2\cos(\theta_1 - \theta_2) = \sqrt{3}^2 = 3$ Therefore: $2 + 2\cos(\theta_1 - \theta_2) = 3$ Solving for $\cos(\theta_1 - \theta_2)$: $\cos(\theta_1 - \theta_2) = \frac{1}{2}$ This means $\theta_1 - \theta_2 = \pm \frac{\pi}{3}$.

4. Next, we calculate $|z_1 - z_2|$. Using a similar identity for the difference of two complex numbers: $|z_1 - z_2|^2 = |z_1|^2 + |z_2|^2 - 2\cdot|z_1|\cdot|z_2|\cdot\cos(\theta_1 - \theta_2)$ Substituting $|z_1| = 1$, $|z_2| = 1$, and $\cos(\theta_1 - \theta_2) = \frac{1}{2}$, we get: $|z_1 - z_2|^2 = 1 + 1 - 2\cdot1\cdot1\cdot\frac{1}{2} = 2 - 1 = 1$ Therefore: $|z_1 - z_2| = 1$

Final Answer:

The value of $|z_1 - z_2|$ is $1$. Hence, the correct choice is:

$\boxed{1}$

Would you like more details or further clarification on any part of this solution?

Here are some related questions for practice:

1. How do you calculate the argument of a complex number?
2. What is the geometric interpretation of the modulus of the sum of two complex numbers?
3. Can $|z_1 - z_2|$ ever be greater than $|z_1 + z_2|$?
4. What happens when $z_1$ and $z_2$ have the same argument?
5. How would the problem change if $|z_1| \neq |z_2|$?

Tip: When solving complex number problems, converting to polar form often simplifies calculations involving moduli and arguments.