Use a sum or difference formula to find the exact value of the trigonometric function. \( \cos 75^{\circ} \) \( \cos 75^{\circ}= \) (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)

To find \( \cos 75^{\circ} \), we can use the cosine sum formula. We know that \( 75^{\circ} = 45^{\circ} + 30^{\circ} \). Applying the cosine sum formula, we have: \[ \cos(75^{\circ}) = \cos(45^{\circ} + 30^{\circ}) = \cos 45^{\circ} \cos 30^{\circ} - \sin 45^{\circ} \sin 30^{\circ} \] Now we can substitute the known values: - \( \cos 45^{\circ} = \frac{\sqrt{2}}{2} \) - \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \) - \( \sin 45^{\circ} = \frac{\sqrt{2}}{2} \) - \( \sin 30^{\circ} = \frac{1}{2} \) So we have: \[ \cos(75^{\circ}) = \left(\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}\right) - \left(\frac{\sqrt{2}}{2} \cdot \frac{1}{2}\right) \] This simplifies to: \[ \cos(75^{\circ}) = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4} \] Thus, the exact value of \( \cos 75^{\circ} \) is \[ \boxed{\frac{\sqrt{6} - \sqrt{2}}{4}}. \]