1. **Problem Statement:** Find the complex conjugate $A^*$ of the expression $$A = e^{i\phi}(e^{i\theta_1} - 1) + e^{i(\theta_1 + \theta_2)}(e^{i\theta_3} - 1).$$ 2. **Recall the rule for complex conjugates:** - The complex conjugate of $e^{ix}$ is $e^{-ix}$. - The conjugate of a sum is the sum of the conjugates. 3. **Apply conjugate to each term:** $$A^* = \overline{e^{i\phi}(e^{i\theta_1} - 1)} + \overline{e^{i(\theta_1 + \theta_2)}(e^{i\theta_3} - 1)}.$$ 4. **Use conjugate properties:** $$A^* = e^{-i\phi}(e^{-i\theta_1} - 1) + e^{-i(\theta_1 + \theta_2)}(e^{-i\theta_3} - 1).$$ 5. **Final answer:** $$\boxed{A^* = e^{-i\phi}(e^{-i\theta_1} - 1) + e^{-i(\theta_1 + \theta_2)}(e^{-i\theta_3} - 1)}.$$