\( 1 \leftarrow \quad \) Find the exact value of \( \sec 75^{\circ}+\sec 125^{\circ}+\sec 255^{\circ}+\sec 305^{\circ} \) \( \sec 75^{\circ}+\sec 125^{\circ}+\sec 255^{\circ}+\sec 305^{\circ}=\square \) (Simplify your answer.)

To evaluate \( \sec 75^{\circ} + \sec 125^{\circ} + \sec 255^{\circ} + \sec 305^{\circ} \), we first find the secant values of each angle. 1. **Calculate \( \sec 75^{\circ} \)**: \[ \sec 75^{\circ} = \frac{1}{\cos 75^{\circ}} = \frac{1}{\frac{\sqrt{6} - \sqrt{2}}{4}} = \frac{4}{\sqrt{6} - \sqrt{2}} = \frac{4(\sqrt{6} + \sqrt{2})}{4} = \sqrt{6} + \sqrt{2} \] 2. **Calculate \( \sec 125^{\circ} \)**: \[ \sec 125^{\circ} = \frac{1}{\cos 125^{\circ}} = \frac{1}{-\cos(180^{\circ} - 125^{\circ})} = -\sec 55^{\circ} = -\frac{1}{\cos 55^{\circ}} = -(\sqrt{6} - \sqrt{2}) \] 3. **Calculate \( \sec 255^{\circ} \)**: \[ \sec 255^{\circ} = \frac{1}{\cos 255^{\circ}} = \frac{1}{-\cos(255^{\circ} - 180^{\circ})} = -\sec 75^{\circ} = -(\sqrt{6} + \sqrt{2}) \] 4. **Calculate \( \sec 305^{\circ} \)**: \[ \sec 305^{\circ} = \frac{1}{\cos 305^{\circ}} = \frac{1}{\cos(360^{\circ} - 305^{\circ})} = \sec 55^{\circ} = \sqrt{6} - \sqrt{2} \] Combine all these results together: \[ \sec 75^{\circ} + \sec 125^{\circ} + \sec 255^{\circ} + \sec 305^{\circ} = (\sqrt{6} + \sqrt{2}) - (\sqrt{6} - \sqrt{2}) - (\sqrt{6} + \sqrt{2}) + (\sqrt{6} - \sqrt{2}) \] Now simplifying that: \[ = \sqrt{6} + \sqrt{2} - \sqrt{6} + \sqrt{2} - \sqrt{6} - \sqrt{2} + \sqrt{6} - \sqrt{2} = 0 \] Thus the final computed value is: \[ \sec 75^{\circ} + \sec 125^{\circ} + \sec 255^{\circ} + \sec 305^{\circ} = \boxed{0} \]